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 Mathematician Mathematicianss These mathematicians influenced the development of set theory and logic:
Georg Cantor
John Venn
George Boole
Augustus DeMorgan
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Georg Cantor
developed set theory
set theory was not initially accepted because it was radically different
set theory today is widely accepted and
1845 -1918
is use used d in man manyy area areas s 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 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
articulated Boole’s algebra of logic
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
developed an algebra of logic (Boolean Algebra)
featured the operators – – – –
1815-1864
and or not nor (exclusive or)
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Augu Au gust stus us De Mo Morg rgan an
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
1806-1871
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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 proper subset of 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
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 is a superset of
| or :
such that (if a condition is true)
| |
the cardinality of a set 11
Set Theory Symbol ∩
intersection union
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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: –
= {2,4,6,8,10}
–
={ |
is a positive even integer <12}
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Set Notation Elements
an element is a member of a set
notation:
Examples: –
–
means “is an element of” means “is not an element of”
{1, 2, 3, 4} 1
A
6
A
2
A
z
A
{x | x is an even number 2
B
9
B
4
B
z
B
10}
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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” 15
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 16
Set Equality
Two sets are 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.
A = {1, 2, 3, 4} A
B and B
B = {1, 4, 2, 3}
A; therefore, A = B and B = A
A = {1, 2, 2, 3, 4, 1, 2} A
B and B
B = {1, 2, 3, 4}
A; therefore, A = B and B = A 17
Cardinality of Sets
Cardinality refers to the number of Cardinality refers elements in a set
A set has a countable number of elements
An set has at least as many elements as the set of
notation: |A|
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Finite Set 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
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
U = 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
the empty set is a subset of every set including itself
notation: { } or
A = {x | x is a Chevrolet Mustang} B = {x | x is a positive number < 0} 22
The Power Set (
)
The power set is the set of all subsets that can be created from a given set
The of the power set is 2 to the power of the given set’s cardinality
notation:
(set name)
A = {a, b, c} where |A| = 3 (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, } and | (A)| = 8
|A| = , then | (A) | = 2 23
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
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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 A = {x | x ε Z+ and x ≤ 9} B = {x | x ε Z+ ; x is even and A
B
B
A
A
B
Elements 123456789 ≤
8} 2 4 6 8
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Example 3 Set Definition
Elements
A = {x | x ε Z+ ; x is even and ≤ 10} 2 4 6 8 10 B = x ε Z+ ; x is odd and x ≤ 10 } A
B
B
A
13579
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Example 4 Set Definition U = 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 = 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 = 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}
= {2, 4, 6, 8, 10}
= (1, 3, 6, 7, 8}
= {3, 7}
(a) Illustrate the sets U, , and in a Venn diagram, marking all the elements in the appropriate places. (b)
, list the elements in each of the following sets: ∩
{6, 8} {1,2, 3, 4, 6, 7, 8, 10}
′
{1, 3, 5, 7, 9}
′
{2, 4, 5, 9, 10}
∩
′
{1, 3, 7}
∩
′
{1, 6, 8}
–
{2, 4, 10}
Δ
{1, 2, 3, 4, 7, 10} 32
Some Properties A
A B and B
A B
A and A B
B
|A B| = |A| |A| + |B| |B| - |A B|
A B
A B
Bc Ac
A B = A Bc
If A B = then we say ‘A’ and ‘B’ are disjoint. 33
Algebra of Sets
Idempotent laws – –
Associative laws
– – 34
Algebra of Sets ctd…
Commutative laws – –
Distributive laws – – 35
Algebra of Sets ctd…
Identity laws – – – –
Involution laws – 36
Algebra of Sets ctd…
Complement laws – – – –
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Algebra of Sets ctd…
De Morgan’s laws – –
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 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 – 39
1_ Proofs 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
( )
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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
(A B)c = Ac Bc ( )
( )
( ) 41
2_ Proofs 2_ Proofs Using Venn Diagrams A 4
A 1
B
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
( )
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Proofs Using Venn Diagrams ctd… Ac : 3, 4 Bc : 1, 4 Ac Bc : 4 (A B)c = Ac Bc ( )
( ) ( )
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Indiana
University Trustees
http://math.comsci.us/sets/index.html
http://library http://library.thinkquest.org/C .thinkquest.org/C0126820/start.html 0126820/start.html
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