Complement (Set Theory)

Complement (set theory)From Wikipedia, the free encyclopedia

Contents

1 Algebra of sets 11.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The fundamental laws of set algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The principle of duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Some additional laws for unions and intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Some additional laws for complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 The algebra of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 The algebra of relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Binary relation 62.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Complement (set theory) 163.1 Relative complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

i

ii CONTENTS

3.2 Absolute complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Complements in various programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Disjoint sets 224.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Disjoint union 265.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Set theory definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Category theory point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Element (mathematics) 286.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Empty set 317.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

CONTENTS iii

7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Family of sets 378.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Finite set 399.1 Definition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3 Necessary and sufficient conditions for finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.4 Foundational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.5 Set-theoretic definitions of finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.5.1 Other concepts of finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10 Intersection (set theory) 4410.1 Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10.1.1 Intersecting and disjoint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2 Arbitrary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3 Nullary intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11 Power set 5011.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.3 Representing subsets as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.4 Relation to binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.6 Subsets of limited cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

iv CONTENTS

11.7 Power object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.8 Functors and quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

12 Subset 5512.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

13 Union (set theory) 5913.1 Union of two sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5913.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.3 Finite unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.4 Arbitrary unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13.4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.4.2 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 1

Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, andcomplementation and the relations of set equality and set inclusion. It also provides systematic procedures for evalu-ating expressions, and performing calculations, involving these operations and relations.Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union,the meet operator being intersection, and the complement operator being set complement.

1.1 Fundamentals

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition andmultiplicationare associative and commutative, so are set union and intersection; just as the arithmetic relation “less than or equal”is reflexive, antisymmetric and transitive, so is the set relation of “subset”.It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations ofequality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory,and for a full rigorous axiomatic treatment see axiomatic set theory.

1.2 The fundamental laws of set algebra

The binary operations of set union ( ∪ ) and intersection ( ∩ ) satisfy many identities. Several of these identities or“laws” have well established names.

Commutative laws:

• A ∪B = B ∪A

• A ∩B = B ∩A

Associative laws:

• (A ∪B) ∪ C = A ∪ (B ∪ C)

• (A ∩B) ∩ C = A ∩ (B ∩ C)

Distributive laws:

• A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)

• A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)

The analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking.Like addition and multiplication, the operations of union and intersection are commutative and associative, and inter-section distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

1

2 CHAPTER 1. ALGEBRA OF SETS

Two additional pairs of laws involve the special sets called the empty set Ø and the universal set U ; together withthe complement operator (AC denotes the complement of A). The empty set has no members, and the universal sethas all possible members (in a particular context).

Identity laws:

• A ∪∅ = A

• A ∩ U = A

Complement laws:

• A ∪AC = U

• A ∩AC = ∅

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Øand U are the identity elements for union and intersection, respectively.Unlike addition and multiplication, union and intersection do not have inverse elements. However the complementlaws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.The preceding five pairs of laws—the commutative, associative, distributive, identity and complement laws—encompassall of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.Note that if the complement laws are weakened to the rule (AC)C = A , then this is exactly the algebra of proposi-tional linear logic.

1.3 The principle of duality

See also: Duality (order theory)

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other byinterchanging ∪ and ∩, and also Ø and U.These are examples of an extremely important and powerful property of set algebra, namely, the principle of dualityfor sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions andintersections, interchanging U and Ø and reversing inclusions is also true. A statement is said to be self-dual if it isequal to its own dual.

1.4 Some additional laws for unions and intersections

The following proposition states six more important laws of set algebra, involving unions and intersections.PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:

idempotent laws:

• A ∪A = A

• A ∩A = A

domination laws:

• A ∪ U = U

• A ∩∅ = ∅

absorption laws:

• A ∪ (A ∩B) = A

• A ∩ (A ∪B) = A

1.5. SOME ADDITIONAL LAWS FOR COMPLEMENTS 3

As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws statedabove. As an illustration, a proof is given below for the idempotent law for union.Proof:

The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law forunion, namely the idempotent law for intersection.Proof:

Intersection can be expressed in terms of set difference :A ∩B = A∖ (A∖B)

1.5 Some additional laws for complements

The following proposition states five more important laws of set algebra, involving complements.PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgan’s laws:

• (A ∪B)C = AC ∩BC

• (A ∩B)C = AC ∪BC

double complement or Involution law:

• (AC)C= A

complement laws for the universal set and the empty set:

• ∅C = U

• UC = ∅

Notice that the double complement law is self-dual.The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies thecomplement laws. In other words, complementation is characterized by the complement laws.PROPOSITION 5: Let A and B be subsets of a universe U, then:

uniqueness of complements:

• If A ∪B = U , and A ∩B = ∅ , then B = AC

1.6 The algebra of inclusion

The following proposition says that inclusion, that is the binary relation of one set being a subset of another, is apartial order.PROPOSITION 6: If A, B and C are sets then the following hold:

reflexivity:

• A ⊆ A

antisymmetry:

• A ⊆ B and B ⊆ A if and only if A = B

transitivity:

4 CHAPTER 1. ALGEBRA OF SETS

• If A ⊆ B and B ⊆ C , then A ⊆ C

The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice, andhence together with the distributive and complement laws above, show that it is a Boolean algebra.PROPOSITION 7: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:

• ∅ ⊆ A ⊆ S

existence of joins:

• A ⊆ A ∪B

• If A ⊆ C and B ⊆ C , then A ∪B ⊆ C

existence of meets:

• A ∩B ⊆ A

• If C ⊆ A and C ⊆ B , then C ⊆ A ∩B

The following proposition says that the statement A ⊆ B is equivalent to various other statements involving unions,intersections and complements.PROPOSITION 8: For any two sets A and B, the following are equivalent:

• A ⊆ B

• A ∩B = A

• A ∪B = B

• A∖B = ∅• BC ⊆ AC

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of setunion or set intersection, which means that the notion of set inclusion is axiomatically superfluous.

1.7 The algebra of relative complements

The following proposition lists several identities concerning relative complements and set-theoretic differences.PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:

• C \ (A ∩B) = (C \A) ∪ (C \B)

• C \ (A ∪B) = (C \A) ∩ (C \B)

• C \ (B \A) = (A ∩ C) ∪ (C \B)

• (B \A) ∩ C = (B ∩ C) \A = B ∩ (C \A)• (B \A) ∪ C = (B ∪ C) \ (A \ C)

• A \A = ∅• ∅ \A = ∅• A \∅ = A

• B \A = AC ∩B

• (B \A)C = A ∪BC

• U \A = AC

• A \ U = ∅

1.8. SEE ALSO 5

1.8 See also• σ-algebra is an algebra of sets, completed to include countably infinite operations.

• Axiomatic set theory

• Field of sets

• Naive set theory

• Set (mathematics)

1.9 References• Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. “TheAlgebra of Sets”, pp 16—23

• Courant, Richard, Herbert Robbins, Ian Stewart,What is mathematics?: An Elementary Approach to Ideas andMethods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. “SUPPLEMENT TO CHAPTER IITHE ALGEBRA OF SETS”

1.10 External links• Operations on Sets at ProvenMath

Chapter 2

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

2.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

6

2.2. SPECIAL TYPES OF BINARY RELATIONS 7

2.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

2.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

2.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

8 CHAPTER 2. BINARY RELATION

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

2.3. RELATIONS OVER A SET 9

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

2.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B→ C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A→ C and g: B → C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

2.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

10 CHAPTER 2. BINARY RELATION

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

2.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

2.4. OPERATIONS ON BINARY RELATIONS 11

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

2.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

2.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

12 CHAPTER 2. BINARY RELATION

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

2.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

2.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

2.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

2.7. EXAMPLES OF COMMON BINARY RELATIONS 13

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

2.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

2.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

2.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

14 CHAPTER 2. BINARY RELATION

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

2.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products and

Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

2.11. EXTERNAL LINKS 15

2.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 3

Complement (set theory)

In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complementof A with respect to a set B is the set of elements in B but not in A. When all sets under consideration are consideredto be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.

3.1 Relative complement

If A and B are sets, then the relative complement of A in B,[1] also termed the set-theoretic difference of B andA,[2] is the set of elements in B, but not in A.

The relative complement of A (left circle) in B (right circle): B ∩Ac = B \A

The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard (sometimes written B – A,but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b – a, where b is taken fromB and a from A).

16

3.2. ABSOLUTE COMPLEMENT 17

Formally

B \A = {x ∈ B |x /∈ A}.

Examples:

• {1,2,3} ∖ {2,3,4} = {1}• {2,3,4} ∖ {1,2,3} = {4}• If R is the set of real numbers and Q is the set of rational numbers, then R \ Q = I is the set ofirrational numbers.

The following lists some notable properties of relative complements in relation to the set-theoretic operations of unionand intersection.If A, B, and C are sets, then the following identities hold:

• C ∖ (A ∩ B) = (C ∖ A)∪(C ∖ B)• C ∖ (A ∪ B) = (C ∖ A)∩(C ∖ B)• C ∖ (B ∖ A) = (C ∩ A)∪(C ∖ B)

[ Alternately written: A ∖ (B ∖ C) = (A ∖ B)∪(A ∩ C) ]

• (B ∖ A) ∩ C = (B ∩ C) ∖ A = B∩(C ∖ A)• (B ∖ A) ∪ C = (B ∪ C) ∖ (A ∖ C)• A ∖ A = Ø• Ø ∖ A = Ø• A ∖ Ø = A

3.2 Absolute complement

If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simplycomplement) of A, and is denoted by Ac or sometimes A′. The same set often[3] is denoted by ∁UA or ∁A if U isfixed, that is:

Ac = U ∖ A.

For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of evennumbers.The following lists some important properties of absolute complements in relation to the set-theoretic operations ofunion and intersection.If A and B are subsets of a universe U, then the following identities hold:

De Morgan’s laws:[1]

• (A ∪B)c= Ac ∩Bc.

• (A ∩B)c= Ac ∪Bc.

Complement laws:[1]

• A ∪Ac = U.

• A ∩Ac = ∅.• ∅c = U.

18 CHAPTER 3. COMPLEMENT (SET THEORY)

The absolute complement of A in U : Ac = U \A

• U c = ∅.• IfA ⊂ B then ,Bc ⊂ Ac.

(this follows from the equivalence of a conditional with its contrapositive)

Involution or double complement law:

• (Ac)c= A.

Relationships between relative and absolute complements:

• A ∖ B = A ∩ Bc

• (A ∖ B)c = Ac ∪ B

The first two complement laws above shows that if A is a non-empty, proper subset of U, then {A, Ac} is a partitionof U.

3.3 Notation

In the LaTeX typesetting language, the command \setminus[4] is usually used for rendering a set difference symbol,which is similar to a backslash symbol. When rendered the \setminus command looks identical to \backslash exceptthat it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. Avariant \smallsetminus is available in the amssymb package.

3.4 Complements in various programming languages

Some programming languages allow for manipulation of sets as data structures, using these operators or functions toconstruct the difference of sets a and b:

3.4. COMPLEMENTS IN VARIOUS PROGRAMMING LANGUAGES 19

.NET Framework a.Except(b);

C++ set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());

Clojure (clojure.set/difference a b)[5]

Common Lisp set-difference, nset-difference[6]

F# Set.difference a b[7]

or

a - b[8]

Falcon diff = a - b[9]

Haskell difference a b

a \\ b[10]

Java diff = a.clone();

diff.removeAll(b);[11]

Julia setdiff[12]

Mathematica Complement[13]

MATLAB setdiff[14]

OCaml Set.S.diff[15]

Octave setdiff[16]

Pascal SetDifference := a - b;

Perl 5 #for perl version >= 5.10

@a = grep {not $_ ~~ @b} @a;

Perl 6 $A ∖ $B

$A (-) $B # texas version

PHP array_diff($a, $b);[17]

Prolog a(X),\+ b(X).

Python diff = a.difference(b)[18]

diff = a - b[18]

R setdiff[19]

Racket (set-subtract a b)[20]

20 CHAPTER 3. COMPLEMENT (SET THEORY)

Ruby diff = a - b[21]

Scala a.diff(b)[22]

or

a -- b[22]

Smalltalk (Pharo) a difference: b

SQL SELECT * FROM A

EXCEPT SELECT * FROM B

Unix shell comm −23 a b[23]

grep -vf b a # less efficient, but works with small unsorted sets

3.5 See also• Algebra of sets

• Naive set theory

• Symmetric difference

3.6 References[1] Halmos (1960) p.17

[2] Devlin (1979) p.6

[3] Bourbaki p. E II.6

[4] The Comprehensive LaTeX Symbol List

[5] clojure.set API reference

[6] Common Lisp HyperSpec, Function set-difference, nset-difference. Accessed on September 8, 2009.

[7] Set.difference<'T> Function (F#). Accessed on July 12, 2015.

[8] Set.( - )<'T> Method (F#). Accessed on July 12, 2015.

[9] Array subtraction, data structures. Accessed on July 28, 2014.

[10] Data.Set (Haskell)

[11] Set (Java 2 Platform SE 5.0). JavaTM 2 Platform Standard Edition 5.0 API Specification, updated in 2004. Accessed onFebruary 13, 2008.

[12] . The Standard Library--Julia Language documentation. Accessed on September 24, 2014

[13] Complement. Mathematica Documentation Center for version 6.0, updated in 2008. Accessed on March 7, 2008.

[14] Setdiff. MATLAB Function Reference for version 7.6, updated in 2008. Accessed on May 19, 2008.

[15] Set.S (OCaml).

[16] . GNU Octave Reference Manual

[17] PHP: array_diff, PHP Manual

3.7. EXTERNAL LINKS 21

[18] . Python v2.7.3 documentation. Accessed on January 17, 2013.

[19] R Reference manual p. 410.

[20] . The Racket Reference. Accessed on May 19, 2015.

[21] Class: Array Ruby Documentation

[22] scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.

[23] comm(1), Unix Seventh Edition Manual, 1979.

• Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van NostrandCompany. Zbl 0087.04403.

• Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN0-387-90441-7. Zbl 0407.04003.

• Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8.

3.7 External links• Weisstein, Eric W., “Complement”, MathWorld.

• Weisstein, Eric W., “Complement Set”, MathWorld.

Chapter 4

Disjoint sets

This article is about the mathematical concept. For the data structure, see Disjoint-set data structure.In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are

A BTwo disjoint sets.

sets whose intersection is the empty set.[1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and{3, 4, 5} are not.

4.1 Generalizations

This definition of disjoint sets can be extended to any family of sets. A family of sets is pairwise disjoint ormutuallydisjoint if every two different sets in the family are disjoint.[1] For example, the collection of sets { {1}, {2}, {3}, ...} is pairwise disjoint.Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two infinite setswhose intersection is a finite set may be said to be almost disjoint.[2]

In topology, there are various notions of separated sets with more strict conditions than disjointness. For instance,two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods. Similarly, ina metric space, positively separated sets are sets separated by a nonzero distance.[3]

22

4.2. EXAMPLES 23

A

BC

A pairwise disjoint family of sets

4.2 Examples

• The set of the drum and the guitar is disjoint to the set of the card and the book

• A pairwise disjoint family of sets

• A non pairwise disjoint family of sets

4.3 Intersections

Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections.

24 CHAPTER 4. DISJOINT SETS

Two sets A and B are disjoint if and only if their intersection A∩B is the empty set.[1] It follows from this definitionthat every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.[4]

A family F of sets is pairwise disjoint if, for every two sets in the family, their intersection is empty.[1] If the familycontains more than one set, this implies that the intersection of the whole family is also empty. However, a familyof only one set is pairwise disjoint, regardless of whether that set is empty, and may have a non-empty intersection.Additionally, a family of sets may have an empty intersection without being pairwise disjoint.[5] For instance, thethree sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not pairwise disjoint. In fact, there are no twodisjoint sets in this collection. Also the empty family of sets is pairwise disjoint.[6]

A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that arepairwise disjoint. For instance, the closed intervals of the real numbers form a Helly family: if a family of closedintervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it mustbe pairwise disjoint.[7]

4.4 Disjoint unions and partitions

A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X.[8] Every partition canequivalently be described by an equivalence relation, a binary relation that describes whether two elements belongto the same set in the partition.[8] Disjoint-set data structures[9] and partition refinement[10] are two techniques incomputer science for efficiently maintaining partitions of a set subject to, respectively, union operations that mergetwo sets or refinement operations that split one set into two.A disjoint union may mean one of two things. Most simply, it may mean the union of sets that are disjoint.[11] Butif two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make themdisjoint before forming the union of the modified sets.[12] For instance two sets may be made disjoint by replacingeach element by an ordered pair of the element and a binary value indicating whether it belongs to the first or secondset.[13] For families of more than two sets, one may similarly replace each element by an ordered pair of the elementand the index of the set that contains it.[14]

4.5 See also• Hyperplane separation theorem for disjoint convex sets

• Mutually exclusive events

• Relatively prime, numbers with disjoint sets of prime divisors

• Set packing, the problem of finding the largest disjoint subfamily of a family of sets

4.6 References[1] Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 15, ISBN 9780387900926.

[2] Halbeisen, Lorenz J. (2011), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer monographs inmathematics, Springer, p. 184, ISBN 9781447121732.

[3] Copson, Edward Thomas (1988),Metric Spaces, Cambridge Tracts in Mathematics 57, Cambridge University Press, p. 62,ISBN 9780521357326.

[4] Oberste-Vorth, Ralph W.; Mouzakitis, Aristides; Lawrence, Bonita A. (2012), Bridge to Abstract Mathematics, MAAtextbooks, Mathematical Association of America, p. 59, ISBN 9780883857793.

[5] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2010), A Transition to Advanced Mathematics, Cengage Learning,p. 95, ISBN 9780495562023.

[6] See answers to the question ″Is the empty family of sets pairwise disjoint?″

[7] Bollobás, Béla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-bridge University Press, p. 82, ISBN 9780521337038.

4.7. EXTERNAL LINKS 25

[8] Halmos (1960), p. 28.

[9] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), “Chapter 21: Data structures forDisjoint Sets”, Introduction to Algorithms (Second ed.), MIT Press, pp. 498–524, ISBN 0-262-03293-7.

[10] Paige, Robert; Tarjan, Robert E. (1987), “Three partition refinement algorithms”, SIAM Journal on Computing 16 (6):973–989, doi:10.1137/0216062, MR 917035.

[11] Ferland, Kevin (2008), Discrete Mathematics: An Introduction to Proofs and Combinatorics, Cengage Learning, p. 45,ISBN 9780618415380.

[12] Arbib, Michael A.; Kfoury, A. J.; Moll, Robert N. (1981), A Basis for Theoretical Computer Science, The AKM series inTheoretical Computer Science: Texts and monographs in computer science, Springer-Verlag, p. 9, ISBN 9783540905738.

[13] Monin, Jean François; Hinchey,Michael Gerard (2003),Understanding FormalMethods, Springer, p. 21, ISBN9781852332471.

[14] Lee, John M. (2010), Introduction to Topological Manifolds, Graduate Texts in Mathematics 202 (2nd ed.), Springer, p.64, ISBN 9781441979407.

4.7 External links• Weisstein, Eric W., “Disjoint Sets”, MathWorld.

Chapter 5

Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation thatindexes the elements according to which set they originated in. Or slightly different from this, the disjoint union ofa family of subsets is the usual union of the subsets which are pairwise disjoint – disjoint sets means they have noelement in common.Note that these two concepts are different but strongly related. Moreover, it seems that they are essentially identicalto each other in category theory. That is, both are realizations of the coproduct of category of sets.

5.1 Example

Disjoint union of sets A0 = {1, 2, 3} and A1 = {1, 2} can be computed by finding:

A∗0 = {(1, 0), (2, 0), (3, 0)}

A∗1 = {(1, 1), (2, 1)}

so

A0 ⊔A1 = A∗0 ∪A∗

1 = {(1, 0), (2, 0), (3, 0), (1, 1), (2, 1)}

5.2 Set theory definition

Formally, let {Ai : i ∈ I} be a family of sets indexed by i. The disjoint union of this family is the set

⊔i∈I

Ai =∪i∈I

{(x, i) : x ∈ Ai}.

The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which Aithe element x came from.Each of the sets Ai is canonically isomorphic to the set

A∗i = {(x, i) : x ∈ Ai}.

Through this isomorphism, one may consider that Ai is canonically embedded in the disjoint union. For i ≠ j, the setsAi* and Aj* are disjoint even if the sets Ai and Aj are not.In the extreme case where each of the Ai is equal to some fixed set A for each i ∈ I, the disjoint union is the Cartesianproduct of A and I:

26

5.3. CATEGORY THEORY POINT OF VIEW 27

⊔i∈I

Ai = A× I.

One may occasionally see the notation

∑i∈I

Ai

for the disjoint union of a family of sets, or the notation A + B for the disjoint union of two sets. This notation ismeant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the termsin the family. Compare this to the notation for the Cartesian product of a family of sets.Disjoint unions are also sometimes written

⊎i∈I Ai or ·

∪i∈I Ai .

In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satisfies theassociated universal property. This also means that the disjoint union is the categorical dual of the Cartesian productconstruction. See coproduct for more details.For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation,the indexed family can be treated simply as a collection of sets. In this case A∗

i is referred to as a copy of Ai and thenotation

∪∗

A∈C

A is sometimes used.

5.3 Category theory point of view

In category theory the disjoint union is defined as a coproduct in the category of sets.As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of thecoproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.This justifies the second definition in the lead.This categorical aspect of the disjoint union explains why

⨿is frequently used, instead of

⊔, to denote coproduct.

5.4 See also• Coproduct

• Disjoint union (topology)

• Disjoint union of graphs

• Partition of a set

• Sum type

• Tagged union

• Union (computer science)

5.5 References• Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised thirded.), New York: Springer-Verlag, p. 60, ISBN 978-0-387-95385-4

• Weisstein, Eric W., “Disjoint Union”, MathWorld.

Chapter 6

Element (mathematics)

In mathematics, an element, ormember, of a set is any one of the distinct objects that make up that set.

6.1 Sets

Writing A = {1, 2, 3, 4} means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A,for example {1, 2}, are subsets of A.Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3,and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colorsred, green and blue.

6.2 Notation and terminology

First usage of the symbol ϵ in the work Arithmetices principia nova methodo exposita by Giuseppe Peano.

The relation “is an element of”, also called set membership, is denoted by the symbol "∈". Writing

x ∈ A

means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A"and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, howeversome authors use them to mean instead "x is a subset of A".[1] Logician George Boolos strongly urged that “contains”be used for membership only and “includes” for the subset relation only.[2]

Another possible notation for the same relation is

28

6.3. CARDINALITY OF SETS 29

A ∋ x,

meaning "A contains x", though it is used less often.The negation of set membership is denoted by the symbol "∉". Writing

x /∈ A

means that "x is not an element of A".The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita. Herehe wrote on page X:

“Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b; ...”

which means

“The symbol ϵ means is. So a ϵ b has to be read as a is a b; ...”

Thereby ϵ is a derivation from the lowercase Greek letter epsilon ("ε") and shall be the first letter of the word ἐστί,which means “is”.The Unicode characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209('not an element of'). The equivalent LaTeX commands are "\in”, "\ni” and "\notin”. Mathematica has commands"\[Element]" and "\[NotElement]".

6.3 Cardinality of sets

Main article: Cardinality

The number of elements in a particular set is a property known as cardinality; informally, this is the size of a set. Inthe above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3. An infiniteset is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The aboveexamples are examples of finite sets. An example of an infinite set is the set of positive integers = { 1, 2, 3, 4, ... }.

6.4 Examples

Using the sets defined above, namely A = {1, 2, 3, 4 }, B = {1, 2, {3, 4}} and C = { red, green, blue }:

• 2 ∈ A

• {3,4} ∈ B

• {3,4} is a member of B

• Yellow ∉ C

• The cardinality of D = { 2, 4, 8, 10, 12 } is finite and equal to 5.

• The cardinality of P = { 2, 3, 5, 7, 11, 13, ...} (the prime numbers) is infinite (this was proven by Euclid).

6.5 References[1] Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. p. 12

[2] George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture). (Speech). Massachusetts Institute of Technology,Cambridge, MA.

30 CHAPTER 6. ELEMENT (MATHEMATICS)

6.6 Further reading• Halmos, Paul R. (1974) [1960], Naive Set Theory, Undergraduate Texts in Mathematics (Hardcover ed.), NY:Springer-Verlag, ISBN 0-387-90092-6 - “Naive” means that it is not fully axiomatized, not that it is silly oreasy (Halmos’s treatment is neither).

• Jech, Thomas (2002), “Set Theory”, Stanford Encyclopedia of Philosophy

• Suppes, Patrick (1972) [1960], Axiomatic Set Theory, NY: Dover Publications, Inc., ISBN 0-486-61630-4 -Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, theaxiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thoroughunderstanding of “set element”.

6.7 External links• Weisstein, Eric W., “Element”, MathWorld.

Chapter 7

Empty set

"∅" redirects here. For similar symbols, see Ø (disambiguation).In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size orcardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for “empty set”, but is now a technical term in measure theory. The empty setmay also be called the void set.

7.1 Notation

Common notations for the empty set include "{}", "∅", and " ∅ ". The latter two symbols were introduced by theBourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets(and not related in any way to the Greek letter Φ).[1]

The empty-set symbol ∅ is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

7.2 Properties

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of “the emptyset” rather than “an empty set”.The mathematical symbols employed below are explained here.For any set A:

• The empty set is a subset of A:

∀A : ∅ ⊆ A

• The union of A with the empty set is A:

∀A : A ∪ ∅ = A

• The intersection of A with the empty set is the empty set:

∀A : A ∩ ∅ = ∅

• The Cartesian product of A and the empty set is the empty set:

∀A : A× ∅ = ∅

31

32 CHAPTER 7. EMPTY SET

The empty set is the set containing no elements.

The empty set has the following properties:

• Its only subset is the empty set itself:

∀A : A ⊆ ∅ ⇒ A = ∅

• The power set of the empty set is the set containing only the empty set:

2∅ = {∅}

7.2. PROPERTIES 33

A symbol for the empty set

• Its number of elements (that is, its cardinality) is zero:

card(∅) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

• For every element of ∅ the property holds (vacuous truth);

• There is no element of ∅ for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

• For every element of V the property holds;

• There is no element of V for which the property holds,

V = ∅

34 CHAPTER 7. EMPTY SET

By the definition of subset, the empty set is a subset of any set A, as every element x of ∅ belongs to A. If it is nottrue that every element of ∅ is in A, there must be at least one element of ∅ that is not present in A. Since there areno elements of ∅ at all, there is no element of ∅ that is not in A. Hence every element of ∅ is in A, and ∅ is a subsetof A. Any statement that begins “for every element of ∅ " is not making any substantive claim; it is a vacuous truth.This is often paraphrased as “everything is true of the elements of the empty set.”

7.2.1 Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

7.3 In other areas of mathematics

7.3.1 Extended real numbers

Since the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two “numbers”or “points” to the real numbers, namely negative infinity, denoted −∞, which is defined to be less than every otherextended real number, and positive infinity, denoted +∞, which is defined to be greater than every other extendedreal number, then:

sup ∅ = min({−∞,+∞} ∪ R) = −∞,

and

inf ∅ = max({−∞,+∞} ∪ R) = +∞.

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound(inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinityis the identity element for the maximum and supremum operators, while positive infinity is the identity element forminimum and infimum.

7.3.2 Topology

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a “clopen” set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every finite set is compact.The closure of the empty set is empty. This is known as “preservation of nullary unions.”

7.3.3 Category theory

If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set isthe unique initial object of the category of sets and functions.

7.4. QUESTIONED EXISTENCE 35

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the emptyset to be open. This empty topological space is the unique initial object in the category of topological spaces withcontinuous maps.The empty set is more ever a strict initial object: only the empty set has a function to the empty set.

7.4 Questioned existence

7.4.1 Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness followsfrom the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

• There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

• In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

7.4.2 Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity,whose meaning and usefulness are debated by philosophers and logicians.The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something.This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explainsthat the empty set is not nothing, but rather “the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.”[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites that the contrast can be seen by rewriting the statements “Nothing is better than eternal happiness” and "[A]ham sandwich is better than nothing” in a mathematical tone. According to Darling, the former is equivalent to “Theset of all things that are better than eternal happiness is ∅ " and the latter to “The set {ham sandwich} is better thanthe set ∅ ". It is noted that the first compares elements of sets, while the second compares the sets themselves.[4]

Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, … we should not assume thatits utility in calculation is dependent upon its actually denoting some object.”

it is also the case that:

“All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers’, in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.”[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantification over individuals, without reifying sets as singular entities having other entities as members.[6]

36 CHAPTER 7. EMPTY SET

7.5 See also• Inhabited set

• Nothing

7.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] • George Boolos, 1984, “To be is to be the value of a variable,” The Journal of Philosophy 91: 430–49. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard Univ. Press: 54–72.

7.7 References• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

• Graham, Malcolm (1975), Modern Elementary Mathematics (HARDCOVER) (in English) (2nd ed.), NewYork: Harcourt Brace Jovanovich, ISBN 0155610392

7.8 External links• Weisstein, Eric W., “Empty Set”, MathWorld.

Chapter 8

Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family ofsubsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.The term “collection” is used here because, in some contexts, a family of sets may be allowed to contain repeatedcopies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

8.1 Examples• The power set P(S) is a family of sets over S.

• The k-subsets S(k) of a set S form a family of sets.

• Let S = {a,b,c,1,2}, an example of a family of sets over S (in the multiset sense) is given by F = {A1, A2, A3,A4} where A1 = {a,b,c}, A2 = {1,2}, A3 = {1,2} and A4 = {a,b,1}.

• The class Ord of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a properclass.

8.2 Special types of set family• A Sperner family is a family of sets in which none of the sets contains any of the others. Sperner’s theorembounds the maximum size of a Sperner family.

• A Helly family is a family of sets such that any minimal subfamily with empty intersection has bounded size.Helly’s theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

8.3 Properties• Any family of subsets of S is itself a subset of the power set P(S) if it has no repeated members.

• Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe).

• Hall’s marriage theorem, due to Philip Hall gives necessary and sufficient conditions for a finite family ofnon-empty sets (repetitions allowed) to have a system of distinct representatives.

8.4 Related concepts

Certain types of objects from other areas ofmathematics are equivalent to families of sets, in that they can be describedpurely as a collection of sets of objects of some type:

37

38 CHAPTER 8. FAMILY OF SETS

• A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges,each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any familyof sets can be interpreted as a hypergraph that has the union of the sets as its vertices.

• An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shapeformed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face.In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family offinite sets without repetitions in which the subsets of any set in the family also belong to the family forms anabstract simplicial complex.

• An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called theincidence relation, specifying which points belong to which lines. An incidence structure can be specified bya family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to eachline, and any family of sets can be interpreted as an incidence structure in this way.

• A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length.When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A blockcode can also be described as a family of sets, by describing each codeword as the set of positions at which itcontains a 1.

8.5 See also• Indexed family

• Class (set theory)

• Combinatorial design

• Russell’s paradox (or Set of sets that do not contain themselves)

8.6 Notes[1] Brualdi 2010, pg. 322

[2] Roberts & Tesman 2009, pg. 692

[3] Biggs 1985, pg. 89

8.7 References• Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0

• Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN0-13-602040-2

• Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN978-1-4200-9982-9

Chapter 9

Finite set

In mathematics, a finite set is a set that has a finite number of elements. For example,

{2, 4, 6, 8, 10}

is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer)and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positiveintegers is infinite:

{1, 2, 3, . . .}.

Finite sets are particularly important in combinatorics, themathematical study of counting. Many arguments involvingfinite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finiteset to a smaller finite set.

9.1 Definition and terminology

Formally, a set S is called finite if there exists a bijection

f : S → {1, . . . , n}

for some natural number n. The number n is the set’s cardinality, denoted as |S|. The empty set {} or Ø is consideredfinite, with cardinality zero.If a set is finite, its elements may be written as a sequence:

S = {x1, x2, . . . , xn}.

In combinatorics, a finite set with n elements is sometimes called an n-set and a subset with k elements is called ak-subset. For example, the set {5,6,7} is a 3-set – a finite set with three elements – and {6,7} is a 2-subset of it.

9.2 Basic properties

Any proper subset of a finite set S is finite and has fewer elements than S itself. As a consequence, there cannot exista bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite. Usingthe standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this requires at least the axiom ofcountable choice.

39

40 CHAPTER 9. FINITE SET

Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection).Similarly, any surjection between two finite sets of the same cardinality is also an injection.The union of two finite sets is finite, with

|S ∪ T | ≤ |S|+ |T |.

In fact:

|S ∪ T | = |S|+ |T | − |S ∩ T |.

More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite,with:

|S × T | = |S| × |T |.

Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with n elements has 2n distinct subsets.That is, the power set of a finite set is finite, with cardinality 2n.Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite.All finite sets are countable, but not all countable sets are finite. (Some authors, however, use “countable” to mean“countably infinite”, so do not consider finite sets to be countable.)The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by setunion.

9.3 Necessary and sufficient conditions for finiteness

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:

1. S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbersless than some specific natural number.

2. (Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with theempty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowskifiniteness.)

3. (Paul Stäckel) S can be given a total ordering which is well-ordered both forwards and backwards. That is,every non-empty subset of S has both a least and a greatest element in the subset.

4. Every one-to-one function from P(P(S)) into itself is onto. That is, the powerset of the powerset of S isDedekind-finite (see below).[1]

5. Every surjective function from P(P(S)) onto itself is one-to-one.

6. (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.[2](Equivalently, every non-empty family of subsets of S has a maximal element with respect to inclusion.)

7. S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderingson S have exactly one order type.

If the axiom of choice is also assumed (the axiom of countable choice is sufficient), then the following conditions areall equivalent:

1. S is a finite set.

2. (Richard Dedekind) Every one-to-one function from S into itself is onto.

3. Every surjective function from S onto itself is one-to-one.

4. S is empty or every partial ordering of S contains a maximal element.

9.4. FOUNDATIONAL ISSUES 41

9.4 Foundational issues

Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus thedistinction between the finite and the infinite lies at the core of set theory. Certain foundationalists, the strict finitists,reject the existence of infinite sets and thus advocate a mathematics based solely on finite sets. Mainstream mathe-maticians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarilyfinite sets constitutes a model of Zermelo–Fraenkel set theory with the axiom of infinity replaced by its negation.Even for thosemathematicians who embrace infinite sets, in certain important contexts, the formal distinction betweenthe finite and the infinite can remain a delicate matter. The difficulty stems from Gödel’s incompleteness theorems.One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice versa), so theincompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. In particular,there exists a plethora of so-called non-standard models of both theories. A seeming paradox, non-standard modelsof the theory of hereditarily finite sets contain infinite sets --- but these infinite sets look finite from within themodel. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.)On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-orderpredicates, can characterize the standard part of all such models. So, at least from the point of view of first-orderlogic, one can only hope to characterize finiteness approximately.More generally, informal notions like set, and particularly finite set, may receive interpretations across a range offormal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories includeZermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theoriesof their various models. One may also choose among classical first-order logic, various higher-order logics andintuitionistic logic.A formalist might see the meaning of set varying from system to system. A Platonist might view particular formalsystems as approximating an underlying reality.

9.5 Set-theoretic definitions of finiteness

In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S asfinite if S admits a bijection to some set of natural numbers of the form {x|x < n} . Mathematicians more typicallychoose to ground notions of number in set theory, for example they might model natural numbers by the order typesof finite well-ordered sets. Such an approach requires a structural definition of finiteness that does not depend onnatural numbers.Interestingly, various properties that single out the finite sets among all sets in the theory ZFC turn out logicallyinequivalent in weaker systems such as ZF or intuitionistic set theories. Two definitions feature prominently in theliterature, one due to Richard Dedekind, the other to Kazimierz Kuratowski. (Kuratowski’s is the definition usedabove.)A set S is called Dedekind infinite if there exists an injective, non-surjective function f : S → S . Such a functionexhibits a bijection between S and a proper subset of S, namely the image of f. Given a Dedekind infinite set S,a function f, and an element x that is not in the image of f, we can form an infinite sequence of distinct elementsof S, namely x, f(x), f(f(x)), ... . Conversely, given a sequence in S consisting of distinct elements x1, x2, x3, ..., we can define a function f such that on elements in the sequence f(xi) = xi+1 and f behaves like the identityfunction otherwise. Thus Dedekind infinite sets contain subsets that correspond bijectively with the natural numbers.Dedekind finite naturally means that every injective self-map is also surjective.Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powersetP(S) with the structure of a semi-lattice. Writing K(S) for the sub-semi-lattice generated by the empty set and thesingletons, call set S Kuratowski finite if S itself belongs to K(S).[3] Intuitively, K(S) consists of the finite subsets of S.Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since onemay obtain K(S) simply by taking the intersection of all sub-semi-lattices containing the empty set and the singletons.Readers unfamiliar with semi-lattices and other notions of abstract algebra may prefer an entirely elementary formu-lation. Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X ofP(S) such that:

• X contains the empty set;

42 CHAPTER 9. FINITE SET

• For every set T in P(S), if X contains T then X also contains the union of T with any singleton.

Then K(S) may be defined as the intersection of M.In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. In the parlance of a popular pedagogical for-mulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose onesock from more than finitely many of the pairs. That would make the set of such socks Dedekind finite: there can beno infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs bychoosing the first sock in the sequence. However, Kuratowski finiteness would fail for the same set of socks.

9.5.1 Other concepts of finiteness

In ZF set theory without the axiom of choice, the following concepts of finiteness for a set S are distinct. They arearranged in strictly decreasing order of strength. In other words, if a set S meets one of the criteria in this list, itmeets all of the criteria which follow that one. In the absence of the axiom of choice, the reverse implications are allunprovable. If the axiom of choice is assumed, then all of these concepts are equivalent.[4] (Note that none of thesedefinitions need the set of finite ordinal numbers to be defined first. They are all pure “set-theoretic” definitions interms of the equality and element-of relations, not involving ω.)

• I-finite. Every non-empty set of subsets of S has a ⊆-maximal element. (This is equivalent to requiring theexistence of a ⊆-minimal element. It is also equivalent to the standard numerical concept of finiteness.)

• Ia-finite. For every partition of S into two sets, at least one of the two sets is I-finite.

• II-finite. Every non-empty ⊆-monotone set of subsets of S has a ⊆-maximal element.

• III-finite. The power set P(S) is Dedekind finite.

• IV-finite. S is Dedekind finite.

• V-finite. ∣S∣ = 0 or 2⋅∣S∣ > ∣S|.

• VI-finite. ∣S∣ = 0 or ∣S∣ = 1 or ∣S∣² > ∣S|.

• VII-finite. S is I-finite or not well-orderable.

The forward implications (from strong to weak) are theorems within ZF. Counter-examples to the reverse implications(from weak to strong) are found using model theory.[5]

Most of these finiteness definitions and their names are attributed to Tarski 1954 by Howard & Rubin 1998, p.278. However, definitions I, II, III, IV and V were presented in Tarski 1924, pp. 49, 93, together with proofs (orreferences to proofs) for the forward implications. At that time, model theory was not sufficiently advanced to findthe counter-examples.

9.6 See also

• FinSet

• Ordinal number

• Peano arithmetic

9.7 Notes[1] The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power

set was shown in 1912 by Whitehead & Russell 2009, p. 288. This Whitehead/Russell theorem is described in moremodern language by Tarski 1924, pp. 73–74.

9.8. REFERENCES 43

[2] Tarski 1924, pp. 48–58, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski’sset-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by Kuratowski1920, pp. 130–131.

[3] The original paper by Kuratowski 1920 defined a set S to be finite when

P(S)∖{∅} = ⋂{X ∈ P(P(S)∖{∅}); (∀x∈S, {x}∈X) and (∀A,B∈X, A∪B∈X)}.

In other words, S is finite when the set of all non-empty subsets of S is equal to the intersection of all classes X whichsatisfy:

• all elements of X are non-empty subsets of S,• the set {x} is an element of X for all x in S,• X is closed under pairwise unions.

Kuratowski showed that this is equivalent to the numerical definition of a finite set.

[4] This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998, pp. 278–280,and Lévy 1958, pp. 2–3, although the details of the presentation of the definitions differ in some respects which do notaffect the meanings of the concepts.

[5] Lévy 1958 found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of theresults to earlier papers by Mostowski and Lindenbaum.

9.8 References• Dedekind, Richard (2012), Was sind und was sollen die Zahlen?, Cambridge Library Collection (Paperbacked.), Cambridge, UK: Cambridge University Press, ISBN 978-1-108-05038-8

• Dedekind, Richard (1963), Essays on the Theory of Numbers, Dover Books on Mathematics, Beman, WoosterWoodruff (Paperback ed.), Dover Publications Inc., ISBN 0-486-21010-3

• Herrlich, Horst (2006), Axiom of Choice, Lecture Notes in Math. 1876, Berlin: Springer-Verlag, ISBN 3-540-30989-6

• Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: Amer-ican Mathematical Society. ISBN 9780821809778.

• Kuratowski, Kazimierz (1920), “Sur la notion d'ensemble fini” (PDF), Fundamenta Mathematicae 1: 129–131

• Lévy, Azriel (1958). “The independence of various definitions of finiteness” (PDF). FundamentaMathematicae46: 1–13.

• Suppes, Patrick (1972) [1960], Axiomatic Set Theory, Dover Books on Mathematics (Paperback ed.), DoverPublications Inc., ISBN 0-486-61630-4

• Tarski, Alfred (1924). “Sur les ensembles finis” (PDF). Fundamenta Mathematicae 6: 45–95.

• Tarski, Alfred (1954). “Theorems on the existence of successors of cardinals, and the axiom of choice”.Nederl. Akad. Wetensch. Proc. Ser. A., Indagationes Math. 16: 26–32. MR 0060555.

• Whitehead, Alfred North; Russell, Bertrand (February 2009) [1912]. Principia Mathematica. Volume Two.Merchant Books. ISBN 978-1-60386-183-0.

9.9 External links• Barile, Margherita, “Finite Set”, MathWorld.

Chapter 10

Intersection (set theory)

Intersection of two sets:A ∩B

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belongto B (or equivalently, all elements of B that also belong to A), but no other elements.[1]

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

10.1 Basic definition

The intersection of A and B is written "A ∩ B". Formally:

A ∩B = {x : x ∈ A ∧ x ∈ B}

44

10.1. BASIC DEFINITION 45

Intersection of three sets:A ∩B ∩ C

that is

x ∈ A ∩ B if and only if

• x ∈ A and• x ∈ B.

For example:

• The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.• The number 9 is not in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the setof odd numbers {1, 3, 5, 7, 9, 11, …}.[2]

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example,is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus,A ∩ (B ∩ C) = (A ∩ B) ∩ C.

46 CHAPTER 10. INTERSECTION (SET THEORY)

Intersections of the Greek, English and Russian alphabet (upper case graphemes)

Inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. Now theintersection of A and Bmay be written as the complement of the union of their complements, derived easily from DeMorgan’s laws:A ∩ B = (Ac ∪ Bc)c

10.1.1 Intersecting and disjoint sets

We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if Aintersects B at some element. A intersects B if their intersection is inhabited.We say that A and B are disjoint if A does not intersect B. In plain language, they have no elements in common. Aand B are disjoint if their intersection is empty, denoted A ∩B = ∅ .For example, the sets {1, 2} and {3, 4} are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6, 12, 18 and other numbers.

10.2. ARBITRARY INTERSECTIONS 47

Example of an intersection with sets

10.2 Arbitrary intersections

The most general notion is the intersection of an arbitrary nonempty collection of sets. IfM is a nonempty set whoseelements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, xis an element of A. In symbols:

(x ∈

∩M)⇔ (∀A ∈ M, x ∈ A) .

The notation for this last concept can vary considerably. Set theorists will sometimes write "⋂M", while others willinstead write "⋂A∈M A". The latter notation can be generalized to "⋂i∈I Ai", which refers to the intersection of thecollection {Ai : i ∈ I}. Here I is a nonempty set, and Ai is a set for every i in I.In the case that the index set I is the set of natural numbers, notation analogous to that of an infinite series may be

48 CHAPTER 10. INTERSECTION (SET THEORY)

seen:

∞∩i=1

Ai.

When formatting is difficult, this can also be written "A1 ∩ A2 ∩ A3 ∩ ...”, even though strictly speaking, A1 ∩ (A2 ∩(A3 ∩ ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for anexample see the article on σ-algebras.)Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it shouldbe of a larger size (⋂).

10.3 Nullary intersection

Conjunctions of the arguments in parenthesesThe conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe.

Note that in the previous section we excluded the case whereM was the empty set (∅). The reason is as follows: Theintersection of the collection M is defined as the set (see set-builder notation)

∩M = {x : ∀A ∈ M, x ∈ A}.

IfM is empty there are no sets A inM, so the question becomes “which x's satisfy the stated condition?" The answerseems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So theintersection of the empty family should be the universal set (the identity element for the operation of intersection) [3]

10.4. SEE ALSO 49

Unfortunately, according to standard (ZFC) set theory, the universal set does not exist. A partial fix for this problemcan be found if we agree to restrict our attention to subsets of a fixed set U called the universe. In this case theintersection of a family of subsets of U can be defined as

∩M = {x ∈ U : ∀A ∈ M, x ∈ A}.

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set byassumption and becomes the identity element for this operation.

10.4 See also• Complement

• Intersection graph

• Logical conjunction

• Naive set theory

• Symmetric difference

• Union

• Cardinality

• Iterated binary operation

• MinHash

10.5 References[1] “Stats: Probability Rules”. People.richland.edu. Retrieved 2012-05-08.

[2] How to find the intersection of sets

[3] Megginson, Robert E. (1998), “Chapter 1”, An introduction to Banach space theory, Graduate Texts in Mathematics 183,New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3

10.6 Further reading• Devlin, K. J. (1993). The Joy of Sets: Fundamentals of Contemporary Set Theory (Second ed.). New York,NY: Springer-Verlag. ISBN 3-540-94094-4.

• Munkres, James R. (2000). “Set Theory and Logic”. Topology (Second ed.). Upper Saddle River: PrenticeHall. ISBN 0-13-181629-2.

• Rosen, Kenneth (2007). “Basic Structures: Sets, Functions, Sequences, and Sums”. Discrete Mathematics andIts Applications (Sixth ed.). Boston: McGraw-Hill. ISBN 978-0-07-322972-0.

10.7 External links• Weisstein, Eric W., “Intersection”, MathWorld.

Chapter 11

Power set

For the search engine developer, see Powerset (company).In mathematics, the power set (or powerset) of any set S, written P(S) , ℘(S), P(S), ℙ(S) or 2S , is the set of all

{x,y,z}

{y,z}{x,z}{x,y}

{y} {z}{x}

Ø

The elements of the power set of the set {x, y, z} ordered in respect to inclusion.

subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFCaxioms), the existence of the power set of any set is postulated by the axiom of power set.[1]

Any subset of P(S) is called a family of sets over S.

11.1 Example

If S is the set {x, y, z}, then the subsets of S are:

50

11.2. PROPERTIES 51

• {} (also denoted ∅ , the empty set)

• {x}

• {y}

• {z}

• {x, y}

• {x, z}

• {y, z}

• {x, y, z}

and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.[2]

11.2 Properties

If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2n . This fact, which is themotivation for the notation 2S , may be demonstrated simply as follows,

We write any subset of S in the format {ω1, ω2, . . . , ωn} where ωi, 1 ≤ i ≤ n , can take the value of 0or 1 . If ωi = 1 , the i -th element of S is in the subset; otherwise, the i -th element is not in the subset.Clearly the number of distinct subsets that can be constructed this way is 2n .

Cantor’s diagonal argument shows that the power set of a set (whether infinite or not) always has strictly highercardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor’stheorem shows that the power set of a countably infinite set is uncountably infinite. For example, the power set of theset of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see cardinality of thecontinuum).The power set of a set S, together with the operations of union, intersection and complement can be viewed as theprototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic tothe Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no longer true, but everyinfinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone’s representationtheorem).The power set of a set S forms an abelian group when considered with the operation of symmetric difference (with theempty set as the identity element and each set being its own inverse) and a commutative monoid when considered withthe operation of intersection. It can hence be shown (by proving the distributive laws) that the power set consideredtogether with both of these operations forms a Boolean ring.

11.3 Representing subsets as functions

In set theory, XY is the set of all functions from Y to X. As “2” can be defined as {0,1} (see natural number), 2S (i.e.,{0,1}S) is the set of all functions from S to {0,1}. By identifying a function in 2S with the corresponding preimageof 1, we see that there is a bijection between 2S and P(S) , where each function is the characteristic function of thesubset in P(S) with which it is identified. Hence 2S and P(S) could be considered identical set-theoretically. (Thusthere are two distinct notational motivations for denoting the power set by 2S : the fact that this function-representationof subsets makes it a special case of the XY notation and the property, mentioned above, that |2S| = 2|S|.)This notion can be applied to the example above in which S = {x, y, z} to see the isomorphism with the binarynumbers from 0 to 2n−1 with n being the number of elements in the set. In S, a 1 in the position corresponding tothe location in the set indicates the presence of the element. So {x, y} = 110.For the whole power set of S we get:

• { } = 000 (Binary) = 0 (Decimal)

52 CHAPTER 11. POWER SET

• {x} = 100 = 4

• {y} = 010 = 2

• {z} = 001 = 1

• {x, y} = 110 = 6

• {x, z} = 101 = 5

• {y, z} = 011 = 3

• {x, y, z} = 111 = 7

11.4 Relation to binomial theorem

The power set is closely related to the binomial theorem. The number of sets with k elements in the power set of aset with n elements will be a combination C(n, k), also called a binomial coefficient.For example the power set of a set with three elements, has:

• C(3, 0) = 1 set with 0 elements

• C(3, 1) = 3 sets with 1 element

• C(3, 2) = 3 sets with 2 elements

• C(3, 3) = 1 set with 3 elements.

11.5 Algorithms

If S is a finite set, there is a recursive algorithm to calculate P(S) .Define the operation F(e, T ) = {X ∪ {e}|X ∈ T}In English, return the set with the element eadded to each set X in T .

• If S = {} ,then P(S) = {{}} is returned.

• Otherwise:

• Let ebe any single element of S .• Let T = S \ {e} , where ' S \ {e} ' denotes the relative complement of {e}in S .• And the result: P(S) = P(T ) ∪ F(e,P(T )) is returned.

In other words, the power set of the empty set is the set containing the empty set and the power set of any other setis all the subsets of the set containing some specific element and all the subsets of the set not containing that specificelement.

11.6 Subsets of limited cardinality

The set of subsets of S of cardinality less than κ is denoted by Pκ(S) or P<κ(S) . Similarly, the set of non-emptysubsets of S might be denoted by P≥1(S) .

11.7. POWER OBJECT 53

11.7 Power object

A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective theidea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structureor algebra.Now the power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and everycomplete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitraryalgebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, andevery algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard subalgebras behaveanalogously to subsets.However there are two important properties of subsets that do not carry over to subalgebras in general. First, althoughthe subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebrasof an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereasthe subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that aclass of algebras contains an algebra that can play the role of 2 in this way.Certain classes of algebras enjoy both of these properties. The first property is more common, the case of havingboth is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, ahomomorphism h: G→ H consists of two functions, one mapping vertices to vertices and the other mapping edges toedges. The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges arerespectively the vertex and edge functions appearing in that set. Furthermore the subgraphs of a multigraph G are inbijection with the graph homomorphisms fromG to the multigraph Ω definable as the complete directed graph on twovertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge,namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph ΩG,called the power object of G.What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts ofelements forming a set V of vertices and E of edges, and has two unary operations s,t: E → V giving the source(start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf.Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Sucha class is a special case of the more general notion of elementary topos as a category that is closed (and moreovercartesian closed) and has an object Ω, called a subobject classifier. Although the term “power object” is sometimesused synonymously with exponential object YX, in topos theory Y is required to be Ω.

11.8 Functors and quantifiers

In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint ofa functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifieris the left adjoint.[3]

11.9 See also• Set theory

• Axiomatic set theory

• Family of sets

• Field of sets

11.10 Notes[1] Devlin (1979) p.50

[2] Puntambekar, A.A. (2007). Theory Of Automata And Formal Languages. Technical Publications. pp. 1–2. ISBN 978-81-8431-193-8.

54 CHAPTER 11. POWER SET

[3] Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 Seepage 58

11.11 References• Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN0-387-90441-7. Zbl 0407.04003.

• Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van NostrandCompany. Zbl 0087.04403.

• Puntambekar, A.A. (2007). Theory Of Automata And Formal Languages. Technical Publications. ISBN978-81-8431-193-8.

11.12 External links• Weisstein, Eric W., “Power Set”, MathWorld.

• Power set at PlanetMath.org.

• Power set in nLab

• Power object in nLab

Chapter 12

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

55

56 CHAPTER 12. SUBSET

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation defines a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

12.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B ,or equivalently

• B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B.

or equivalently

• B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S).When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.[1]

12.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may ormay not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ isproper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

12.3 Examples

• The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true.

• The set D = {1, 2, 3} is a subset of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false).

• Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

• The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any setexcept itself.

• The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}

• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a linesegment is a proper subset of the set of points in a line. These are two examples in which both the subset andthe whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is,the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.

• The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinitebut the latter set has a larger cardinality (or power) than the former set.

12.4. OTHER PROPERTIES OF INCLUSION 57

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but no proper subset of B

12.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identifiedwith the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumeratingS = {s1, s2, …, sk} and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from{0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

12.5 See also• Containment order

12.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

58 CHAPTER 12. SUBSET

C B A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

12.7 External links• Weisstein, Eric W., “Subset”, MathWorld.

Chapter 13

Union (set theory)

Union of two sets:A ∪B

In set theory, the union (denoted by ∪) of a collection of sets is the set of all distinct elements in the collection.[1] Itis one of the fundamental operations through which sets can be combined and related to each other.

13.1 Union of two sets

The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In symbols,

A ∪B = {x : x ∈ A or x ∈ B}

59

60 CHAPTER 13. UNION (SET THEORY)

Union of three sets:A ∪B ∪ C

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example(involving two infinite sets) is:

A = {x is an even integer larger than 1}B = {x is an odd integer larger than 1}A ∪B = {2, 3, 4, 5, 6, . . . }

Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multipleoccurrences of identical elements have no effect on the cardinality of a set or its contents.The number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of evennumbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.

13.2 Algebraic properties

Binary union is an associative operation; that is,

13.3. FINITE UNIONS 61

A ∪ (B ∪ C) = (A ∪ B) ∪ C.

The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e., either ofthe above can be expressed equivalently as A ∪ B ∪ C). Similarly, union is commutative, so the sets can be written inany order.The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A, for any set A.These facts follow from analogous facts about logical disjunction.

13.3 Finite unions

One can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains allelements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if andonly if x is in at least one of A, B, and C.In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the unionset is a finite set.

13.4 Arbitrary unions

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. IfM isa set whose elements are themselves sets, then x is an element of the union ofM if and only if there is at least oneelement A ofM such that x is an element of A. In symbols:

x ∈∪

M ⇐⇒ ∃A ∈ M, x ∈ A.

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union inaxiomatic set theory.This idea subsumes the preceding sections, in that (for example) A ∪ B ∪ C is the union of the collection {A,B,C}.Also, if M is the empty collection, then the union of M is the empty set. The analogy between finite unions andlogical disjunction extends to one between arbitrary unions and existential quantification.

13.4.1 Notations

The notation for the general concept can vary considerably. For a finite union of sets S1, S2, S3, . . . , Sn one oftenwrites S1 ∪ S2 ∪ S3 ∪ · · · ∪ Sn . Various common notations for arbitrary unions include

∪M ,

∪A∈MA , and∪

i∈I Ai , the last of which refers to the union of the collection {Ai : i ∈ I} where I is an index set and Ai is a setfor every i ∈ I . In the case that the index set I is the set of natural numbers, one uses a notation

∪∞i=1 Ai analogous

to that of the infinite series. When formatting is difficult, this can also be written "A1 ∪ A2 ∪ A3 ∪ ···". (This lastexample, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras.)Whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.

13.4.2 Union and intersection

Since sets with unions and intersections form a Boolean algebra, Intersection distributes over union:

A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)

and union distributes over intersection:

A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)

62 CHAPTER 13. UNION (SET THEORY)

Within a given universal set, union can be written in terms of the operations of intersection and complement as

A ∪B =(AC ∩BC

)Cwhere the superscript C denotes the complement with respect to the universal set.Arbitrary union and intersection also satisfy the law

∪i∈I

( ∩j∈J

Ai,j

)⊆

∩j∈J

(∪i∈I

Ai,j

)

13.5 See also• Alternation (formal language theory), the union of sets of strings

• Cardinality

• Complement (set theory)

• Disjoint union

• Intersection (set theory)

• Iterated binary operation

• Naive set theory

• Symmetric difference

13.6 Notes[1] Weisstein, Eric W. “Union”. Wolfram’s Mathworld. Retrieved 2009-07-14.

13.7 External links• Weisstein, Eric W., “Union”, MathWorld.

• Hazewinkel, Michiel, ed. (2001), “Union of sets”, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4

• Infinite Union and Intersection at ProvenMath DeMorgan’s laws formally proven from the axioms of set theory.

13.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 63

13.8 Text and image sources, contributors, and licenses

13.8.1 Text• Algebra of sets Source: https://en.wikipedia.org/wiki/Algebra_of_sets?oldid=660214875 Contributors: The Anome, AugPi, Charles

Matthews, WhisperToMe, Wile E. Heresiarch, Tobias Bergemann, Macrakis, Doshell, Discospinster, Esse~enwiki, Slipstream, Paul Au-gust, Oleg Alexandrov, Woohookitty, Isnow, Salix alba, Juan Marquez, Mathbot, Splintercellguy, Trovatore, Arthur Rubin, Gilliam,Bluebot, Javalenok, Byelf2007, Jackzhp, Kupirijo, Josephpetty100, Policron, The enemies of god, Alex10023, Jamelan, Tcamps42, An-chor Link Bot, Hans Adler, Addbot, Legobot, TaBOT-zerem, Matt Popat, Bluerasberry, Corruptcopper, Saeidpourbabak, Yahia.barie,Specs112, MegaSloth, DASHBot, Set theorist, ClueBot NG, MerlIwBot, DonMTobin, Helpful Pixie Bot, Daviddwd, Palltrast, Chris-Gualtieri, Freeze S, Jochen Burghardt, YiFeiBot, Eniacpx, Plcarmelbiron and Anonymous: 41

• Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=669144544 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Complement (set theory) Source: https://en.wikipedia.org/wiki/Complement_(set_theory)?oldid=672482880 Contributors: DamianYerrick, AxelBoldt, Fnielsen, Toby Bartels, Michael Hardy, Wshun, 6birc, Julesd, AugPi, Charles Matthews, Dcoetzee, Wik, E23~enwiki,Hyacinth, Robbot, Peak, Bkell, Anthony, Tobias Bergemann, Giftlite, Lethe, Tweenk, Elektron, Sam Hocevar, TheObtuseAngleOf-Doom, Running, Rich Farmbrough, Paul August, Robertbowerman, , EmilJ, Nk, Mroach, Caesura, Oleg Alexandrov, Mindma-trix, Shreevatsa, Isnow, Salix alba, Juan Marquez, Brighterorange, Tardis, BMF81, Chobot, YurikBot, RussBot, Trovatore, Nathan11g,Haginile~enwiki, Tomisti, Arthur Rubin, Mike1024, SmackBot, Melchoir, Deon Steyn, BiT, Bluebot, Georgelulu, Octahedron80, DHN-bot~enwiki, Acepectif, Pilotguy, Newone, CBM, Pjvpjv, Escarbot, JAnDbot, Kcho, VoABot II, Greg Ward, Gja822, JCraw, Commons-Delinker, Pharaoh of the Wizards, Numbo3, Ttwo, Pasixxxx, Kyle the bot, Hqb, Philmac, AlleborgoBot, Haiviet~enwiki, BotMultichill,Jerryobject, Yoda of Borg, Anchor Link Bot, DEMcAdams, ClueBot, Dead10ck, Ideal gas equation, Alexbot, Watchduck, Ewger, Addbot,Luckas-bot, Yobot, Nallimbot, AnomieBOT, Ciphers, Jim1138, Txebixev, Martnym, RibotBOT, Entropeter, Thehelpfulbot, FrescoBot,Paine Ellsworth, Pinethicket, TobeBot, Lotusoneny, Whisky drinker, 1qaz-pl, EmausBot, AsceticRose, ZéroBot, Alpha Quadrant (alt),Nxtfari, Minoru-kun, ClueBot NG, Bgcaus, MelbourneStar, Pzrq, Catch2011, Chmarkine, Brad7777, Bakkedal, Freeze S, Deltahedron,Lambda Fairy, Manman2323, Gpendergast, Bradleyscollins and Anonymous: 78

• Disjoint sets Source: https://en.wikipedia.org/wiki/Disjoint_sets?oldid=655510316 Contributors: AxelBoldt, Mav, Tarquin, Jeronimo,Shd~enwiki, Arvindn, Toby Bartels, Michael Hardy, Wshun, Revolver, Charles Matthews, Fibonacci, Robbot, Sheskar~enwiki, TobiasBergemann, Giftlite, Fropuff, Achituv~enwiki, Almit39, Nickj, Jumbuck, Dirac1933, Salix alba, FlaBot, Jameshfisher, Chobot, Alge-braist, YurikBot, RobotE, Bota47, Light current, Raijinili, SmackBot, RDBury,Maksim-e~enwiki, DHN-bot~enwiki, Vina-iwbot~enwiki,Typinaway, Mets501, Iridescent, Devourer09, Ezrakilty, Christian75, Kilva, RobHar, Salgueiro~enwiki, JAnDbot, David Eppstein,Miker70741, J.delanoy, STBotD, VolkovBot, Jamelan, PanagosTheOther, Alexbot, DumZiBoT, WikHead, Addbot, LaaknorBot, Sp-Bot, Legobot, Luckas-bot, Yobot, GrouchoBot, Erik9bot, MastiBot, Peacedance, Tbhotch, Ripchip Bot, ZéroBot, 28bot, ClueBot NG,Prakhar.agrwl, Helpful Pixie Bot, BG19bot, Amyxz, JYBot, Stephan Kulla, Domez99, Flat Out and Anonymous: 34

• Disjoint union Source: https://en.wikipedia.org/wiki/Disjoint_union?oldid=664597191 Contributors: Mav, Zundark, Tarquin, Takuya-Murata, Charles Matthews, Dcoetzee, Altenmann, MathMartin, Giftlite, Fropuff, CyborgTosser, Abdull, Paul August, Oleg Alexandrov,Linas, Jacobolus, Salix alba, Mathbot, YurikBot, Hairy Dude, Bhny, Tong~enwiki, David Pierce, SmackBot, RDBury, Melchoir, AlanMc-Beth, Nbarth, Wdvorak, Mets501, Thijs!bot, JAnDbot, Albmont, Americanhero, Pavel Jelínek, The enemies of god, PatHayes, Jamelan,Niceguyedc, Addbot, Arketyp, Gtgith, Vorbeigehende, Yodigo, Erik9bot, Stpasha, WBielas, WikitanvirBot, D.Lazard, ChuispastonBot,Akseli.palen, 10k, Freeze S, Deltahedron, Kfitzell29 and Anonymous: 19

• Element (mathematics) Source: https://en.wikipedia.org/wiki/Element_(mathematics)?oldid=666614530 Contributors: Michael Hardy,Charles Matthews, Hyacinth, Chuunen Baka, Tobias Bergemann, Giftlite, Dbenbenn, Maximaximax, Eep², Brianjd, Dissipate, Paul Au-gust, Elwikipedista~enwiki, Nickj, Mlm42, Oleg Alexandrov, Mutford, Rjwilmsi, Salix alba, Chobot, YurikBot, RobotE, Thane, Goffrie,InverseHypercube, Mhss, Octahedron80, Gracenotes, Khoikhoi, MichaelBillington, Wvbailey, Bjankuloski06en~enwiki, SpyMagician,16@r, LimWei Quan, Mets501, JohnCD, Gregbard, Cydebot, Epbr123, Enlil2, Zorro CX, KeypadSDM, Ttwo, Idioma-bot, Pleasantville,SieBot, Paolo.dL, Harry~enwiki, ClueBot, Phileasson, Heckledpie, SoxBot III, RMFan1, Addbot, Professor Calculus, Betterusername,Binary TSO, Ben Ben, Yobot, TaBOT-zerem, KamikazeBot, Ciphers, Zach2231, Materialscientist, Citation bot, ArthurBot, DSisyphBot,FrescoBot, AllCluesKey, Spartan S58, Pinethicket, Jauhienij, Jophos, Rsagira58, EmausBot, Solarra, Traxs7, Ebrambot, Chuispaston-Bot, ClueBot NG, Wcherowi, SusikMkr, Atomician, Wikih101, Funnydiamond99, Aciganj, YFdyh-bot, Saehry, Stephan Kulla, Jwall42,Sonicbethesame and Anonymous: 78

• Empty set Source: https://en.wikipedia.org/wiki/Empty_set?oldid=666003197 Contributors: AxelBoldt, Lee Daniel Crocker, Uriyan,Bryan Derksen, Tarquin, Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Ryguasu, Hephaestos, Patrick,Michael Hardy,MartinHarper, TakuyaMurata, Eric119, Den fjättrade ankan~enwiki, Andres, Evercat, Renamed user 4, CharlesMatthews,Berteun, Dcoetzee, David Latapie, Dysprosia, Jitse Niesen, Krithin, Hyacinth, Spikey, Jeanmichel~enwiki, Flockmeal, Phil Boswell,

64 CHAPTER 13. UNION (SET THEORY)

Robbot, Sanders muc, Peak, Romanm, Gandalf61, Henrygb, Wikibot, Pengo, Tobias Bergemann, Adam78, Tosha, Giftlite, Dbenbenn,Vfp15, BenFrantzDale, Herbee, Fropuff, MichaelHaeckel, Macrakis, Python eggs, Rdsmith4, Mike Rosoft, Brianjd, Mormegil, Guan-abot, Paul August, Spearhead, EmilJ, BrokenSegue, Nortexoid, 3mta3, Obradovic Goran, Jonathunder, ABCD, Sligocki, Dzhim, Itsmine,HenryLi, Hq3473, Angr, Isnow, Qwertyus, MarSch, Salix alba, Bubba73, ChongDae, Salvatore Ingala, Chobot, YurikBot, RussBot,Rsrikanth05, Trovatore, Ms2ger, Saric, EtherealPurple, GrinBot~enwiki, TomMorris, SmackBot, InverseHypercube, Melchoir, FlashSh-eridan, Ohnoitsjamie, Joefaust, SMP, J. Spencer, Octahedron80, Iit bpd1962, Tamfang, Cybercobra, Dreadstar, RandomP, Jon Awbrey,Jóna Þórunn, Lambiam, Jim.belk, Vanished user v8n3489h3tkjnsdkq30u3f, Loadmaster, Hvn0413, Mets501, EdC~enwiki, Joseph Solisin Australia, Spindled, James pic, Amalas, Philiprbrenan, CBM, Gregbard, Cydebot, Pais, Julian Mendez, Malleus Fatuorum, Epbr123,Nick Number, Escarbot, Sluzzelin, .anacondabot, David Eppstein, Ttwo, Maurice Carbonaro, Ian.thomson, It Is Me Here, Daniel5Ko,NewEnglandYankee, DavidCBryant, VolkovBot, Zanardm, Rei-bot, Anonymous Dissident, Andy Dingley, SieBot, Niv.sarig, ToePeu.bot,Randomblue, Niceguyedc, Wounder, Nosolution182, Versus22, Palnot, AmeliaElizabeth, Feinoha, American Eagle, ThisIsMyWikipedi-aName, LaaknorBot, AnnaFrance, Numbo3-bot, Zorrobot, Legobot, Luckas-bot, Yobot, Ciphers, Xqbot, Nasnema, , GrouchoBot,LucienBOT, Pinethicket, Kiefer.Wolfowitz, Abductive, Jauhienij, FoxBot, Lotje, LilyKitty, Woodsy dong peep, EmausBot, Sharlack-Hames, Ystory, ClueBot NG, Cntras, Rezabot, Helpful Pixie Bot, Michael.croghan, Langing, Ugncreative Usergname, JYBot, Kephir,Phinumu, Noyster, GeoffreyT2000, Skw27 and Anonymous: 82

• Family of sets Source: https://en.wikipedia.org/wiki/Family_of_sets?oldid=659156521 Contributors: Toby Bartels, Charles Matthews,Chris Howard, Oleg Alexandrov, Salix alba, Chobot, Wavelength, Arthur Rubin, Reedy, Mhss, CBM, RomanXNS, David Eppstein,JoergenB, Pomte, PixelBot, Avoided, Addbot, Matěj Grabovský, Calle, Erik9bot, DivineAlpha, NearSetAccount, Xnn, Sheerun, ClueBotNG, Wcherowi and Anonymous: 8

• Finite set Source: https://en.wikipedia.org/wiki/Finite_set?oldid=670818425Contributors: AxelBoldt, BryanDerksen, Zundark, James~enwiki,XJaM, Toby Bartels, Edward, Michael Hardy, Rl, Revolver, Charles Matthews, Dysprosia, Hyacinth, Aleph4, Robbot, Tobias Bergemann,Giftlite, Dbenbenn, Smjg, Siroxo, LiDaobing, CSTAR, Mike Rosoft, Guanabot, EmilJ, Emhoo~enwiki, Arthena, Salix alba, VKok-ielov, Mathbot, RexNL, Chobot, YurikBot, Trovatore, Mistercow, Kompik, Arthur Rubin, Sardanaphalus, SmackBot, Reedy, Tsca.bot,Jim.belk, Merkey88, Rschwieb, Fell Collar, JRSpriggs, CRGreathouse, CBM, Gregbard, Sam Staton, Thijs!bot, Escarbot, JAnDbot,Bongwarrior, David Eppstein, Gwern, Ttwo, Alan U. Kennington, VolkovBot, Popopp, Tamorlan, EmxBot, SieBot, Paolo.dL, Classi-calecon, Fsmoura, ClueBot, Cliff, Tomas e, Alexbot, Clayt85, Hans Adler, BodhisattvaBot, Ronhjones, Jasper Deng, Tide rolls, ,مانيLegobot, Luckas-bot, Yobot, TaBOT-zerem, Ciphers, Xqbot, Ssola, Snyderxc, FrescoBot, Trappist the monk, Raiden09, DexDor, Johnof Reading, Mz7, Empty Buffer, Ludovica1, Mjbmrbot, ClueBot NG, Wcherowi, Misshamid, Thatoneguywhoreallylikesxkcd, Trickster-Wolf, ChrisGualtieri, DavidLeighEllis and Anonymous: 42

• Intersection (set theory) Source: https://en.wikipedia.org/wiki/Intersection_(set_theory)?oldid=672268008 Contributors: AxelBoldt,Tarquin, Andre Engels, Toby~enwiki, Toby Bartels, Michael Hardy, Wshun, Booyabazooka, Den fjättrade ankan~enwiki, AugPi, Dpol,Andres, Hashar, Revolver, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Hyacinth, Donarreiskoffer, Robbot, Romanm, Bkell, Millosh,Tobias Bergemann, Giftlite, Fropuff, Sam Hocevar, Creidieki, Mormegil, Mindspillage, Rich Farmbrough, Hydrox, Dbachmann, PaulAugust, Bender235, Elwikipedista~enwiki, EurekaLott, Haham hanuka, Jumbuck, Inky, Bookandcoffee, Kenyon, Mindmatrix, Apokrif,Isnow, Mandarax, Graham87, Salix alba, Chobot, YurikBot, ML, Ms2ger, Bodo Thiesen, Reedy, InverseHypercube, Melchoir, Eskimbot,Object01, Gilliam, Rmosler2100, Octahedron80, DHN-bot~enwiki, Antonrojo, Esproj, Cybercobra, Mets501, EdC~enwiki, Newone,Tophtucker, Devourer09, CBM, Mhaitham.shammaa, Pirtu, Spencer, Gökhan, JaGa, CommonsDelinker, Ttwo, VolkovBot, AnonymousDissident, PaulTanenbaum, Synthebot, AlleborgoBot, SieBot, ToePeu.bot, Pyth~enwiki, Alexbot, Watchduck, Computer97, Bigoperm,Beroal, Addbot, Loupeter, Luckas-bot, Yobot, KamikazeBot, Hizkey, AnomieBOT, Clark89, Jsorr, TobeBot, DARTH SIDIOUS 2,Shafaet, EmausBot, ZéroBot, Chewings72, ChuispastonBot, ClueBot NG, Brad7777, Glacialfox, Lucifer bobby, Stephan Kulla, Brirush,YiFeiBot, Andrybak and Anonymous: 65

• Power set Source: https://en.wikipedia.org/wiki/Power_set?oldid=655854839Contributors: AxelBoldt, Zundark, Tarquin, Awaterl, BoleslavBobcik, Michael Hardy, Wshun, TakuyaMurata, GTBacchus, Pcb21, Mxn, Charles Matthews, Berteun, Dysprosia, Jay, Hyacinth, Ed g2s,.mau., Aleph4, Robbot, Tobias Bergemann, Adam78, Giftlite, Dratman, DefLog~enwiki, Tomruen, Mormegil, Rich Farmbrough, Ted-Pavlic, Paul August, Zaslav, Elwikipedista~enwiki, Spayrard, SgtThroat, Obradovic Goran, Jumbuck, Kocio, Tony Sidaway, Ultramarine,Kenyon, Oleg Alexandrov, Linas, Flamingspinach, GregorB, Yurik, Salix alba, FlaBot, VKokielov, SchuminWeb, Small potato, Cia-Pan, NevilleDNZ, Chobot, YurikBot, Stephenb, Trovatore, Bota47, Deville, Closedmouth, MathsIsFun, Realkyhick, GrinBot~enwiki,SmackBot, InverseHypercube, Persian Poet Gal, SMP, Alink, Octahedron80, Kostmo, Armend, Shdwfeather, 16@r, Mike Fikes, Malter,Freelance Intellectual, JRSpriggs, Vaughan Pratt, CBM, Gregbard, Sam Staton, Goldencako, DumbBOT, Cj67, Abu-Fool Danyal ibnAmir al-Makhiri, Felix C. Stegerman, David Eppstein, R'n'B, RJASE1, UnicornTapestry, VolkovBot, Camrn86, Anonymous Dissident,PaulTanenbaum, Dmcq, AlleborgoBot, Pcruce, Faradayplank, MiNombreDeGuerra, Megaloxantha, KrustallosIce28, S2000magician,Classicalecon, Dmitry Dzhus, PipepBot, DragonBot, He7d3r, Marc van Leeuwen, Addbot, Freakmighty, Download, Luckas-bot, Yobot,Ht686rg90, ArthurBot, La Mejor Ratonera, FrescoBot, Showgun45, ComputScientist, Throw it in the Fire, Tkuvho, HRoestBot, El-Lutzo, John of Reading, WikitanvirBot, Lunaibis, Set theorist, Josve05a, AMenteLibera, Wcherowi, Helpful Pixie Bot, Sebastien.noir,Deltahedron, QuantumNico, Mark viking, Jadiker, Rajiv1965 and Anonymous: 87

• Subset Source: https://en.wikipedia.org/wiki/Subset?oldid=671828052 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,

13.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 65

AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17, Johnkennethcfamero and Anonymous: 183

• Union (set theory) Source: https://en.wikipedia.org/wiki/Union_(set_theory)?oldid=672268577 Contributors: Damian Yerrick, Axel-Boldt, Tarquin, Toby~enwiki, Toby Bartels, Ubiquity, Wshun, Sannse, Eric119, Den fjättrade ankan~enwiki, AugPi, Ideyal, CharlesMatthews, Dcoetzee, Dysprosia, Hyacinth, Sabbut, Donarreiskoffer, Robbot, R3m0t, Romanm, Bkell, Tobias Bergemann, Giftlite, Frop-uff, Macrakis, Tomruen, Sam Hocevar, Mormegil, Jisatsusha, Rich Farmbrough, Paul August, Elwikipedista~enwiki, *drew, Helix84,Petdance, Jumbuck, Bookandcoffee, Kenyon, Oleg Alexandrov, Isnow, Marudubshinki, Jshadias, Salix alba, FlaBot, Chobot, Yurik-Bot, Michael Slone, Trovatore, Bodo Thiesen, Adam majewski, Reedy, Melchoir, Octahedron80, Mets501, EdC~enwiki, Newone, TwasNow, Myncknm, JForget, CmdrObot, Escarbot, Ahecht, Rhalden, JCraw, Ttwo, Coolg49964, Policron, TXiKiBoT, Anonymous Dis-sident, Madhero88, AlleborgoBot, Haiviet~enwiki, SieBot, Gerakibot, Radon210, Thehotelambush, ClueBot, PipepBot, Watchduck,Computer97, Thingg, Addbot, Betterusername, Cesiumfrog, Loupeter, Zorrobot, Jarble, Legobot, Luckas-bot, Yobot, AnakngAraw,AnomieBOT, Tryptofish, Ciphers, Plastichandle, Materialscientist, Jubileeclipman, Xhaoz, RandomDSdevel, Pinethicket, HRoestBot,Hamtechperson, Jauhienij, TobeBot, Saikrrishna.ch, Dinamik-bot, Specs112, EmausBot, ZéroBot, Coasterlover1994, Chewings72, Chuis-pastonBot, Ringil92, ClueBot NG, Jack Greenmaven, Joel B. Lewis, Theopolisme, Skybirdnomad, Brad7777, Minsbot, Jethro B, StephanKulla, Wlter, Dalangster and Anonymous: 93

13.8.2 Images• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi-

nal artist: ?• File:Disjuct-sets.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/89/Disjuct-sets.svg License: CCBY-SA 3.0 Contrib-

utors: Own work Original artist: Svjo• File:Disjunkte_Mengen.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Disjunkte_Mengen.svg License: CC BY

3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)• File:Empty_set.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Empty_set.svg License: Public domain Contribu-

tors: Own work Original artist: Octahedron80• File:First_usage_of_the_symbol_∈.png Source: https://upload.wikimedia.org/wikipedia/commons/f/f4/First_usage_of_the_symbol_

%E2%88%88.png License: Public domainContributors: Arithmetices principia novamethodo exposita. (page X)Original artist: GiuseppePeano

• File:Graph_of_non-injective,_non-surjective_function_(red)_and_of_bijective_function_(green).gif Source: https://upload.wikimedia.org/wikipedia/commons/b/b0/Graph_of_non-injective%2C_non-surjective_function_%28red%29_and_of_bijective_function_%28green%29.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt

• File:Hasse_diagram_of_powerset_of_3.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ea/Hasse_diagram_of_powerset_of_3.svg License: CC-BY-SA-3.0 Contributors: self-made using graphviz's dot. Original artist: KSmrq

• File:Multigrade_operator_AND.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/74/Multigrade_operator_AND.svgLicense: Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Nullset.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Nullset.svg License: CC BY-SA 3.0 Contributors: Ownwork Original artist: Hugo Férée

• File:PolygonsSetIntersection.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5a/PolygonsSetIntersection.svgLicense:CC BY-SA 3.0 Contributors: Own work Original artist: kismalac

• File:PolygonsSet_EN.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/PolygonsSet_EN.svg License: CC0 Contrib-utors: File:PolygonsSet.svg Original artist: File:PolygonsSet.svg: kismalac

• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Subset_with_expansion.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Subset_with_expansion.svg License:CC-BY-SA-3.0 Contributors: Own work Original artist: User:J.Spudeman~{}commonswiki

• File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

• File:Venn0001.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/99/Venn0001.svg License: Public domain Contributors:? Original artist: ?

• File:Venn0010.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5a/Venn0010.svg License: Public domain Contributors:? Original artist: ?

• File:Venn0111.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/30/Venn0111.svg License: Public domain Contributors:? Original artist: ?

• File:Venn1010.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/eb/Venn1010.svg License: Public domain Contributors:? Original artist: ?

• File:Venn_0000_0001.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Venn_0000_0001.svg License: Public do-main Contributors: ? Original artist: ?

• File:Venn_0111_1111.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ee/Venn_0111_1111.svg License: Public do-main Contributors: ? Original artist: ?

66 CHAPTER 13. UNION (SET THEORY)

• File:Venn_A_intersect_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg License: Pub-lic domain Contributors: Own work Original artist: Cepheus

• File:Venn_A_subset_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b0/Venn_A_subset_B.svg License: Public do-main Contributors: Own work Original artist: User:Booyabazooka

• File:Venn_diagram_gr_la_ru.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/Venn_diagram_gr_la_ru.svgLicense:Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Wiki_letter_w_cropped.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg License:CC-BY-SA-3.0 Contributors:

• Wiki_letter_w.svg Original artist: Wiki_letter_w.svg: Jarkko Piiroinen• File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Public

domain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs),based on original logo tossed together by Brion Vibber

13.8.3 Content license• Creative Commons Attribution-Share Alike 3.0