Sets and Subsets Set A set is a collection of well-defined objects (elements/members). The elements of the set are said to belong to (or be contained in) the set. A set may be itself be an element of some other set. A set can be a set of sets of sets and so on. Sets will be denoted be capital letters A, B, C, ... Elements will be denoted by lower case letters a, b, c, .... x, y, z. The phrase “is an element of” will be denoted by the symbol . xA denotes x is an element of the set A. xA denotes x is not an element of the set.
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Sets and Subsets Set A set is a collection of well-defined objects (elements/members). The elements of the set are said to belong to (or be contained in)
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Sets and Subsets
Set
A set is a collection of well-defined objects (elements/members).
The elements of the set are said to belong to (or be contained in) the
set.
A set may be itself be an element of some other set.
A set can be a set of sets of sets and so on.
Sets will be denoted be capital letters A, B, C, ...
Elements will be denoted by lower case letters a, b, c, .... x, y, z.
The phrase “is an element of” will be denoted by the symbol .
xA denotes x is an element of the set A.
xA denotes x is not an element of the set.
Properties of Binary Relations
Equivalence Relations
Compatibility and Partial Ordering Relations Hasse Diagrams
Functions Inverse Functions Composition of functions Recursive Functions
Lattice and its properties
Pigeon hole principle and its application
There are five ways to describe a set.
1. Describe a set by describing the properties of the
members of the set.
2. Describe a set by listing its elements.
3. Describe a set by its characteristic functions defined as
A (x) = 1, if xA
= 0, if xA.
4. Describe a set by a recursive formula.
5. Describe a set by an operation (such as union,
intersection, complement etc.,) on some other sets.
Example: Describe the set containing all the non
negative integers less than or equal to 5.
Let A denote the set.
1. A = {x | x is a non-negative integer 5}
2. A = {0, 1, 2, 3, 4, 5}.
3. A (x) = 1 for x = 0, 1, 2, 3, 4, 5
= 0 for x>5 and x<0 (otherwise)
4. A = {xi+1 = xi +1, i = 0, 1, 2, 3, 4 where x0 = 0}
5. Let B = {0, 2, 4} and C = {1, 3, 5}. A = BC.
Subset: Set A is said to be a subset of set B if every element of set A is an element of B. It is denoted as AB. (A is contained in B)
Ex:
1. If A = {0, 2, 4} and B = {0, 1, 2, 3, 4, 5}, then AB.
2. If A = {0, 1, 2} and B = {0, 1, 2} then AB.
Proper subset: A is said to be a proper subset of B if A is a subset of B and there is at least one element of B which is not in A. It is denoted as AB (A is strictly contained in B).
Ex: If A = {0, 2, 4} and B = {0, 1, 2, 3, 4, 5}, then AB.
Properties:
Let A, B, C be the sets.
1. AA.
2. If AB and BC, then AC.
3. If AB and BC, then AC.
4. If AB and AC, then BC (B is not contained in C).
AB and BA if and only if (iff) A and B have the same elements.
Two sets A and B are equal iff AB and BA. It is denoted as A = B.
To show that two sets A and B are equal, we must show that each
element of A is also an element of B, and conversely.
Empty/Null Set: A set containing no elements is called the
empty/null set and is denoted as .
Example: = { }
The empty set is a subset of every set. i.e., A for every A.
Singleton: A set containing a single element is called a Singleton.
Operations on Sets.There are three operations on sets namely Complement, Union and Intersection.
Complement:Absolute complement: Let U be the universal set and let A be any subset of U. The absolute complement of A, denoted as ’, is defined as {x | x A} or {x | x U and x A}.
Ex:
If U = {0, 1, 2, 3, 4, 5} and A = {0, 2, 4} then A` = {1, 3, 5}.
Relative complement: If A and B are sets, the relative complement of A with respect to B isB – A = {x | x B and x A}.
Ex: If A = {0, 2, 4} and B = {0, 1, 2, 3} then B – A = {1, 3}.
It is clear that `= U and U` = .
The complement of complement of A is equal to A i.e. (A)`` = A.
Ex: If U = {0, 1, 2, 3, 4, 5} and A = {0, 2, 4,}
then A` = {1, 3, 5} and A`` = {0, 2, 4} = A.
Union: The union, denoted as , two sets A and B is A B = {x | x A or x B or x both A and B}.
Ex: If A = {0, 3, 6, 9} and B = {0, 2, 4, 6, 8} then AB = {0, 2, 3, 4, 6, 8, 9}.
Intersection: The intersection, denoted as , of two sets A and B is
A B = {x | x A and x B}.
Ex: If A = {0, 3, 6, 9} and B = {0, 2, 4, 6, 8} then AB = {0, 6}
Basic properties of Union and Intersection:
Union IntersectionIdempotent: AA = A AA = ACommutative: AB = BA AB = BAAssociative: A(BC) = (AB)C (AB)C = A(BC)
Prove or Disprove (AB) C A(BC).
Ex: If A = {0, 1, 2}, B = {1, 3, 4} and C = {2, 4, 6}
Cartesian product: The Cartesian product, denoted as X, of the sets A and B is the set of all ordered pairs (x, y) where xA and yB
i.e., A X B = {(x, y)|xA and yB}.
Ex: Let A = {0, 1} and B = {a, b} A X B = {(0, a), (0, b), (1, a), (1, b)} and B X A = {(a, 0), (a, 1), (b, 0), (b, 1)}.
An ordered pair in a set is specified by two elements in a prescribed order i.e., (a, b) (b, a).
Distributive and DeMorgan’s Laws
Distributive Laws
Let A, B and C be three sets.
A(BC) = (AB)(AC) andA(BC) = (AB) (AC)
Ex: A = {0, 1, 2}, B = {1, 2, 5} and C = {0, 5, 8}BC = {0, 1, 2, 5, 8}AB = {1, 2}AC = {0}
L.H.S = A(BC)= {0, 1, 2}
R.H.S =(AB)(AC)= {0, 1, 2}
A(BC) = (AB)(AC)
A = {0, 1, 2}, B = {1, 2, 5} and C = {0, 5, 8}
BC = {5}AB = {0, 1, 2, 5}AC = {0, 1, 2, 5, 8}
L.H.S = A(BC)= {0, 1, 2, 5}
R.H.S = (AB)(AC)= {0, 1, 2, 5}
ABC) = (AB)(AC)
DeMorgan’s Laws:
Let A and B be two sets.
(AB)` = A`B` and (AB)` = A`B `
Ex: Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8}, A = {0, 1, 2} and B = {1, 2, 5}.
AB = {0, 1, 2, 5}
A` = {3, 4, 5, 6, 7, 8}
B` = {0, 3, 4, 6, 7, 8}
L.H.S = (AB)` = {3, 4, 6, 7, 8}
R.H.S = A`B` = {3, 4, 6, 7, 8}
(AB)` = A`B`
Ex: Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8}, A = {0, 1, 2} and B = {1, 2, 5}.
AB = {1, 2}
A` = {3, 4, 5, 6, 7, 8}B` = {0, 3, 4, 6, 7, 8}
L.H.S = (AB)` = {0, 3, 4, 5, 6, 7, 8}
R.H.S = A`B` = {0, 3, 4, 5, 6, 7, 8}
(AB)` = A`B`
Venn diagrams: Venn diagrams are used to visualize various properties of the set operations.
The Universal Set is represented by a large rectangular area.
U
Subsets are represented by circular areas
A
Venn diagrams of set operations
B
Properties of Binary Relations
Relations
Let A and B be nonempty sets. a relation R A X B and (a, b) R.a is related to b by R.a R b.
Examples:Let A = {1, 2, 3} and B = {r, s}. R = {(1, r), (2, s), (3, r)} is a relation from A to B.
Let A = {1, 2, 3, 4, 5}. Define the relation R (less than) on A: a R b if and only if a < b.R = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}.
Domain of R
Dom(R)Let R X X Y be a relation from X to Y.Set of elements in X those are related to some element in Y. Subset of X.Set of all first elements in the pairs that make up R.
Range of R
Ran(R)Set of elements in Y that are related to some element in X.Subset of Y.Set of all second elements in the pairs that make up R.
The Digraph of a relationGeometrical representations of relations.Example:Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}. The Digraph of R is
1 3
4
2
1.Let A = {1, 2, 3, 4, 5, 6}. Construct pictorial descriptions of the relation R on A for the following:
a. R = {(j, k) | j divides k}b. R = {(j, k) | j is a multiple of k}c. R = {(j, k) | (j – k)2 A}d. R = {(j, k) | j/k is a prime}
2. Let R be the relation from A = {1, 2, 3, 4, 5} to B = {1, 3, 5} which is defined by “x is less than y”. Write as a set of ordered pairs.
3. Find the domain and range of the relation R.A = {a, b, c, d}, B = {1, 2, 3}, R = {(a, 1), (a, 2), (b, 1), (c, 2), (d, 1)}.
4. Find the domain, range and digraph of the relation R.A = {1, 2, 3, 4, 8} = B; a R b if and only if a + b 9.
5. Find the relation determined by the digraph.
6. Find the relation determined by the digraph.
1
4 2
3
5
5
1
2
3
4
Properties of Relations
1.Reflexive
A relation R on a set A is reflexive if (a, a) R for all a A.a R a for all a A.Every element a A is related to itself.Matrix must have all 1’s on its main diagonal.Dom(R) = Ran(R) = A.
Example:Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 2), (2,3), (3, 3)}.
2.Irreflexive
A relation R on a set A is irreflexive if a R a for every a A.No element is related to itself.Matrix must have all 0’s on its main diagonal.
Example:Let A = {1, 2, 3} and R = {(1, 2), (2, 1), (3, 1), (3, 2)}.
Example:Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (3, 1), (3, 2)} Neither reflexive nor irreflexive.
3.SymmetricA relation R on a set A is symmetric if whenever a R b, then b R a.
Example:Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2)}.
4.AsymmetricA relation R on a set A is asymmetric if whenever a R b, then b R a.
Example:Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 2), (3, 2)}.
5. AntisymmetricA relation R on a set A is antisymmetric if whenever a R b and b
R a, then a = b.
Example:Let A = {1, 2, 3}and R = {(1, 1), (2, 2), (3, 3)}.
6. TransitiveA relation R on a set A is transitive if whenever a R b and b R c,
then a R c.
Example:Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.
Exercises:
Let A = {1, 2, 3, 4}. Determine whether the relation is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive.
A relation R in a set A is called an equivalence relation if it is reflexive , symmetric , and transitive.
If R is an equivalence relation in a set X, then D(R), the domain of R isX itself.
Example:Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4, 3), (4, 4)}.
• Equality of numbers on a set of real numbers.• Equality of subsets of a universal set.• Similarity of triangles on the set of triangles.• Relation of statements being equivalent in the set of
statements.
Ex 1): Let X = {1,2,3,4} and
R = {(1,1),(1,4),(4,1),(4,4),(2,2),2,3),(3,2),(3,3)}
Write the matrix of R and sketch its graph.
Ex 2): Let X = {1,2,……..,7} and
R = {(x , y) I x - y is divisible by 3}
Show that R is an equivalence relation .Draw the
graph of R.
Exercises: Determine whether the relation R on the set A is an
Let A = {1, 2, 3, 4, 12}. a. Define the relation R by aRb iff a | b. b. Prove that R is a partial order on A. c. Draw the Hasse diagram.a. R = {(a, b) | (a, b) A and a | b}
= {(1, 1), (1, 2), (1, 3), (1, 4), (1, 12), (2, 2), (2, 4), (2, 12), (3, 3), (3, 12), (4, 4), (4, 12), (12, 12)}b. R is Reflexive since (a, a) R for all a A. R is Transitive since if (a, b) R and (b, c) R, then (a, c) R. R is antisymmetric since if a | b and b | a, then a = b, for all a, b A. R is a partial order on A.c. Hasse Diagram
Upper bound of a subset B of A Element a A, if x R a for all x B.
Lower bound of a subset B of A Element a A, if a R x for all x B.
Least Upper Bound (LUB) / Supremum (Sup) of a subset B of A Element a A, if
A is an upper bound of B.A’ is an upper bound of B, then a R a’.
Greatest Lower Bound (GLB) / Infimum (Inf) of a subset B of A Element a A, if
A is a lower bound of B.A’ is a lower bound of B, then a’ R a.
Given A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {3, 4, 5}, Find, if they exist, all upper bounds of B; all lower bounds of B; the least upper bound of B; and the greatest lower bound of B for the poset given in Figure below.
Upper bounds 6, 7, 8 Lower bounds 1, 2, 3 Least Upper Bound No Greatest Lower Bound 3
Find, if they exist, all upper bounds of B; all lower bounds of B; the least upper bound of B; and the greatest lower bound of B for the poset given below for B = {c, d, e}.
Upper bounds are f, g, h.Lower bounds are a, b, c.LUB (B) = fGLB (B) = c
Lattice and its properties
LatticePOSet (A, R) Every two-element subset {a, b} of A has a Least
Upper Bound (LUB) and Greatest Lower Bound (GLB) in A.
LUB {a, b} a b a join b GLB {a, b} a b a meet bExamples(Z+, |).(Dn, |), Dn be the Set of all positive divisors of n.D20 = {1, 2, 4, 5, 10, 20}(P(S), ).
Sublattice:
Let (L, R) be a lattice and M be a subset of L. Then M is called a sub lattice of L if a b M and a b M whenever a M and b M.
Product of Lattices:
Consider the lattices (L1, R) and (L2, R). Then these are posets. Also, (L1 X L2, R) is a poset under the product partial order defined by
(a, b) R (a`, b`) if a R a` in L1 and b R b` in L2.
L1 X L2 is called the product of L1 and L2.
Special types of Lattices
Bounded Lattice: The least and greatest elements of a lattice are called the
bounds of the lattice and are denoted by 0 and 1.
A lattice which has both elements 0 and 1 is called a bounded lattice.
Complete Lattice: A lattice is called complete if each of its nonempty
subsets has a least upper bound and a greatest lower bound.
Every finite lattice must be complete. Every complete lattice must have a least element and a greatest element.
Complement:
In a bounded lattice an element b L is called a complement of an element a L if
a b = 0 and a b = 1
Complemented Lattice:
A lattice is said to be a complemented lattice if every element of L has at least one complement.
Distributive Lattice:
A lattice is called a distributive lattice for any a,b,c L
a (b c) = (a b) (a c)
a (b c) = (a b) (a c)
Properties of Lattices
Let (L, R) be a Lattice. For every a, b, c L,
Idempotent Properties a a = a a a = aCommutative Properties a b = b a a b = b aAssociative Properties a (b c) = (a b) c a (b c) = (a b) cAbsorption Properties a (a b) = a a (a b) = a a b = b if and only if a R b. a b = a if and only if a R b. a b = a if and only if a b = b.