Set Theory Project

Set Theory

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Understanding set theory helps people to :

see things in terms of systems

organize things into groups

begin to understand logic

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Key Mathematicians

These mathematicians influenced the development of set theory and logic:

Georg Cantor

John Venn

George Boole

Augustus DeMorgan

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Georg Cantor 1845 -1918

developed set theory

set theory was not initially accepted because it was radically different

set theory today is widely accepted and is used in many areas of mathematics

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Cantor the concept of infinity was expanded by

Cantor’s set theory

Cantor proved there are “levels of infinity”

an infinitude of integers initially ending with ωor

an infinitude of real numbers exist between 1 and 2;

there are more real numbers than there are integers…

0ℵ

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John Venn 1834-1923

studied and taught logic and probability theory

articulated Boole’s algebra of logic

devised a simple way to diagram set operations (Venn Diagrams)

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George Boole 1815-1864

self-taught mathematician with an interest in logic

developed an algebra of logic (Boolean Algebra)

featured the operators– and– or– not– nor (exclusive or)

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Augustus De Morgan 1806-1871

developed two laws of negation

interested, like other mathematicians, in using mathematics to demonstrate logic

furthered Boole’s work of incorporating logic and mathematics

formally stated the laws of set theory

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Basic Set Theory Definitions

A set is a collection of elements

An element is an object contained in a set

If every element of Set A is also contained in Set B, then Set A is a subset of Set B– A is a proper subset of B if B has more elements

than A does

The universal set contains all of the elements relevant to a given discussion

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Set Theory SymbolSymbol Meaning

Upper case designates set name

Lower case designates set elements

{ } enclose elements in set

∈ or is (or is not) an element of

⊆ is a subset of (includes equal sets)

⊂ is a proper subset of

⊄ is not a subset of

⊃ is a superset of

| or : such that (if a condition is true)

| | the cardinality of a set

∉

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Set Theory SymbolSymbol Meaning

∩ intersection

∪ union

A or A the compliment of A”; all elements not in AA – B all elements in A but not in Bn(A) the number of elements in A A = B (A is equal to B )A and B contain the sameA ≅ B (A is equivalent to B)

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Set Notation: Defining Sets

a set is a collection of objects

sets can be defined two ways:– by listing each element– by defining the rules for membership

Examples:– A = {2,4,6,8,10}

– A = {x | x is a positive even integer <12}

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Set Notation Elements an element is a member of a set

notation: ∈ means “is an element of”∉ means “is not an element of”

Examples:

– A = {1, 2, 3, 4}

1 ∈ A 6 ∉ A

2 ∈ A z ∉ A

– B = {x | x is an even number ≤ 10}

2 ∈ B 9 ∉ B

4 ∈ B z ∉ B14

Subsets

a subset exists when a set’s members are also contained in another set

notation:

⊆ means “is a subset of”

⊂ means “is a proper subset of”

⊄ means “is not a subset of”

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Subset Relationships A = {x | x is a positive integer ≤ 8}

set A contains: 1, 2, 3, 4, 5, 6, 7, 8

B = {x | x is a positive even integer < 10}

set B contains: 2, 4, 6, 8

C = {2, 4, 6, 8, 10}

set C contains: 2, 4, 6, 8, 10

Subset Relationships

A ⊆ A A ⊄ B A ⊄ C

B ⊂ A B ⊆ B B ⊂ C

C ⊄ A C ⊄ B C ⊆ C

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Set Equality Two sets are equal if and only if they contain precisely

the same elements.

The order in which the elements are listed is unimportant.

Elements may be repeated in set definitions without increasing the size of the sets.

Examples:A = {1, 2, 3, 4} B = {1, 4, 2, 3}

A ⊂ B and B ⊂ A; therefore, A = B and B = A

A = {1, 2, 2, 3, 4, 1, 2} B = {1, 2, 3, 4}

A ⊂ B and B ⊂ A; therefore, A = B and B = A

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Cardinality of Sets

Cardinality refers to the number of elements in a set

A finite set has a countable number of elements

An infinite set has at least as many elements as the set of natural numbers

notation: |A| represents the cardinality of Set A

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Finite Set Cardinality

Set Definition Cardinality

A = {x | x is a lower case letter} |A| = 26

B = {2, 3, 4, 5, 6, 7} |B| = 6

C = {x | x is an even number < 10} |C|= 4

D = {x | x is an even number ≤ 10} |D| = 5

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Infinite Set CardinalitySet Definition Cardinality

A = {1, 2, 3, …} |A| =

B = {x | x is a point on a line} |B| =

C = {x| x is a point in a plane} |C| =

0ℵ

0ℵ

1ℵ

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Universal Sets

The universal set is the set of all things pertinent to a given discussionand is designated by the symbol U

Example:U = {all students at IUPUI}

Some Subsets:

A = {all Computer Technology students}

B = {freshmen students}

C = {sophomore students}

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The Empty Set

Any set that contains no elements is called the empty set

the empty set is a subset of every set including itself

notation: { } or φ

Examples ~ both A and B are emptyA = {x | x is a Chevrolet Mustang}

B = {x | x is a positive number < 0}

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The Power Set ( P ) The power set is the set of all subsets that can

be created from a given set

The cardinality of the power set is 2 to the power of the given set’s cardinality

notation: P (set name)Example:A = {a, b, c} where |A| = 3P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ}

and |P (A)| = 8

In general, if |A| = n, then |P (A) | = 2n

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Special Sets

Z represents the set of integers – Z+ is the set of positive integers and

– Z- is the set of negative integers

N represents the set of natural numbers

ℝ represents the set of real numbers

Q represents the set of rational numbers

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Venn Diagrams

Venn diagrams show relationships between sets and their elements

Universal Set

Sets A & B

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Example 1

Set Definition ElementsA = {x | x ε Z+ and x ≤ 8} 1 2 3 4 5 6 7 8

B = {x | x ε Z+; x is even and ≤ 10} 2 4 6 8 10

A ⊄ B

B ⊄ A

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Example 2

Set Definition ElementsA = {x | x ε Z+ and x ≤ 9} 1 2 3 4 5 6 7 8 9

B = {x | x ε Z+ ; x is even and ≤ 8} 2 4 6 8

A ⊄ B

B ⊂ A

A ⊃ B

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Example 3

Set Definition ElementsA = {x | x ε Z+ ; x is even and ≤ 10} 2 4 6 8 10

B = x ε Z+ ; x is odd and x ≤ 10 } 1 3 5 7 9

A ⊄ B

B ⊄ A

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Example 4Set Definition

U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}

B = {2, 3, 4, 7}

C = {4, 5, 6, 7}

A = {1, 2, 6, 7}

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Example 5

Set DefinitionU = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}

B = {2, 3, 4, 7}

C = {4, 5, 6, 7}

B = {2, 3, 4, 7}

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Example 6

Set DefinitionU = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}

B = {2, 3, 4, 7}

C = {4, 5, 6, 7}

C = {4, 5, 6, 7}

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Operations On Sets Example

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If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 6, 8, 10}

B = (1, 3, 6, 7, 8}

C = {3, 7}

(a) Illustrate the sets U, A, B and C in a Venn diagram, marking all the elements in the appropriate places.

(b) Using your Venn diagram, list the elements in each of the following sets:

A ∩ B = {6, 8}

A ∪ B = {1,2, 3, 4, 6, 7, 8, 10}

A′ = {1, 3, 5, 7, 9}

B′ = {2, 4, 5, 9, 10}

B ∩ A′ = {1, 3, 7}

B ∩ C′ = {1, 6, 8}

A – B = {2, 4, 10}

A Δ B = {1, 2, 3, 4, 7, 10}

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Some Properties

A ⊆ A∪B and B ⊆ A∪B

A∩B ⊆ A and A∩B ⊆ B

|A∪B| = |A| + |B| - |A∩B|

A⊆B ⇒ Bc⊆Ac

A B = A∩Bc

If A∩B = Φ then we say ‘A’ and ‘B’ are disjoint.

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Algebra of Sets

Idempotent laws

–A ∪ A = A–A ∩ A = A

Associative laws

–(A ∪ B) ∪ C = A ∪ (B ∪ C)–(A ∩ B) ∩ C = A ∩ (B ∩ C)

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Algebra of Sets ctd…

Commutative laws

–A ∪ B = B ∪ A–A ∩ B = B ∩ A

Distributive laws

–A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)–A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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Algebra of Sets ctd…

Identity laws

–A ∪ Φ = A–A ∩ U = A–A ∪ U = U–A ∩ Φ = Φ

Involution laws

–(Ac)c = A

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Algebra of Sets ctd…

Complement laws

–A ∪ Ac = U–A ∩ Ac = Φ–Uc = Φ–Φc = U

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Algebra of Sets ctd…

De Morgan’s laws

–(A ∪ B)c = Ac ∩ Bc

–(A ∩ B)c = Ac ∪ Bc

Note: Compare these De Morgan’s laws with the De Morgan’s laws that you find in logic and see the similarity.

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Proofs (example)

Basically there are two approaches in proving above mentioned laws and any other set relationship :

1_ Algebraic method2_ Using Venn diagrams

For example lets discuss how to prove

–(A ∪ B)c = Ac ∩ B c

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1_Proofs Using Algebraic Method

x∈(A∪B)c ⇒ x∉A∪B

⇒ x∉A ∧ x∉B

⇒ x∈Ac ∧ x∈Bc

⇒ x∈Ac∩Bc

⇒ (A∪B)c ⊆ Ac∩Bc (α)

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Proofs Using Algebraic Method ctd…

x∈Ac∩Bc⇒ x∈Ac ∧ x∈Bc

⇒ x∉A ∧ x∉B

⇒ x∉A∪B

⇒ x∈(A∪B)c

⇒ Ac∩Bc ⊆ (A∪B)c (β)

(α) ∧ (β)⇒ (A∪B)c = Ac∩Bc

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2_ Proofs Using Venn Diagrams

Note that these indicated numbers are not the actual members of each set. They are region numbers.

BA

A ∪ B

3

4

1 2

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Proofs Using Venn Diagrams ctd…

U : 1, 2, 3, 4

A : 1, 2 (i.e. The region for ‘A’ is 1 and 2)

B : 2, 3

∴ A∪B : 1, 2, 3

∴ (A∪B)c : 4 (α)

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Proofs Using Venn Diagrams ctd…

Ac : 3, 4

Bc : 1, 4

∴ Ac∩Bc : 4 (β)

(α) ∧ (β)⇒ (A∪B)c = Ac∩Bc

Indiana University Trustees

http://math.comsci.us/sets/index.html

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http://library.thinkquest.org/C0126820/start.html

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