Set Theory Project

Set Theory

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Understanding set theory helps people to : 

see things in terms of systems

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organize things into groups

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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  

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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 

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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 ω ℵ0 or an infinitude of real numbers exist between 1 and 2; there are more real numbers than there are integers… 6

John Venn 

studied and taught logic and probability theory

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articulated Boole’s algebra of logic

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

1834-1923

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George Boole 

self-taught mathematician with an interest in logic

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developed an algebra of logic (Boolean Algebra)

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featured the operators – – – –

1815-1864

and or not nor (exclusive or)

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Augustus De Morgan 

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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 9

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 Symbol Symbol

Meaning

Upper case

designates set name

Lower case

designates set elements

{ } ∈ or

enclose elements in set

∉

is (or is not) an element of

⊆

is a subset of (includes equal sets)

⊂

is a proper subset of

⊄

is not a subset of

⊃ | or :

is a superset of such that (if a condition is true)

| |

the cardinality of a set 11

Set Theory Symbol Symbol ∩ ∪

Meaning intersection union

A or A A–B n(A) A=B

the compliment of A”; all elements not in A all elements in A but not in B the number of elements in A (A is equal to B )A and B contain the same

A≅B

(A is equivalent to B)

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Set Notation: Defining Sets 

a set is a collection of objects

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

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Examples: – A = {2,4,6,8,10} – A = {x | x is a positive even integer <12} 13

Set Notation Elements  

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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∉B

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Subsets 

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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” 15

Subset Relationships 

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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 16

Set Equality 

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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 17

Cardinality of Sets 

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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 19

Infinite Set Cardinality Set Definition

Cardinality

A = {1, 2, 3, …}

|A| =

ℵ

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

|B| =

ℵ

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

|C| =

0

0

ℵ1

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Universal Sets 

The universal set is the set of all things pertinent to a given discussion and 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} 21

The Empty Set 

Any set that contains no elements is called the

empty set 

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the empty set is a subset of every set including itself notation: { } or φ

Examples ~ both A and B are empty A = {x | x is a Chevrolet Mustang} B = {x | x is a positive number < 0} 22

The Power Set ( P ) 

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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| = 3 P (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 23

Special Sets 

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

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N represents the set of natural numbers

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ℝ represents the set of real numbers

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Q represents the set of rational numbers 24

Venn Diagrams 

Venn diagrams show relationships between sets and their elements Sets A & B

Universal Set

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Example 1 Set Definition

Elements

A = {x | x ε Z+ and x ≤ 8} 12345678 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

Elements

A = {x | x ε Z+ and x ≤ 9} 123456789 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

Elements

A = {x | x ε Z+ ; x is even and ≤ 10} 2 4 6 8 10 13579 B = x ε Z+ ; x is odd and x ≤ 10 } A⊄B B⊄A

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Example 4 Set 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 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} B = {2, 3, 4, 7}

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Example 6 Set 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} C = {4, 5, 6, 7} 31

Operations On Sets Example 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 ∪ A c = U –A ∩ A c = Φ –U c = 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 method 2_ Using Venn diagrams 

For example lets discuss how to prove – (A ∪ B) c = Ac ∩ B c 39

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)

⇒ (A∪B)c = Ac∩Bc (α) ∧ (β) 41

2_ Proofs Using Venn Diagrams A∪ B A

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4 1

2

3

B

Note that these indicated numbers are not the actual members of each set. They are region numbers. 42

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)

⇒ (A∪B)c = Ac∩Bc (α) ∧ (β)

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Indiana University Trustees http://math.comsci.us/sets/index.html http://library.thinkquest.org/C0126820/start.html

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N.N.M

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