Discrete Structures
Chapter 6: Set Theory
6.1 Set Theory: Definitions and the Element Method of Proof
Erickson
The introduction of suitable abstractions is our only mental aid to organize and master complexity.
– E. W. Dijkstra, 1930 – 2002
6.1 Set Theory - Definitions and the
Element Method of Proof
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Subsets
• Let’s write what it means for a set
A to be a subset of a set B as a formal universal conditional statement:
A

B
  x , if x

A then x

B .
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Subsets
•
The negation is existential
A B
  x , if x

A and x

B .
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Subsets
•
A proper subset of a set is a subset that is not equal to its containing set.
A is a proper subset of B

1.
A

B , and
2. there is at least one element in B that is not in A .
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Element Argument
•
Let sets X and Y be given. To prove that X

Y ,
1. Suppose that x is a particular but arbitrarily chosen element of X ,
2. Show that x is an element of Y .
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Example – pg. 350 # 4
•
Let A = { n

| n = 5 r for some integer r } and B = { m

| m = 20 s for some integer s }.
a. Is A

B ? Explain. b. Is B

A ? Explain.
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Set Equality
•
Given sets A and B , A equals B , written A = B , iff every element of A is in B and every element of B is in A . Symbolically,
A = B

A

B and B

A
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6.1 Set Theory - Definitions and the
Element Method of Proof
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Operations on Sets
•
Let A and B be subsets of a universal set U .
1. The union of A and B denoted A B , is the set of all elements that are in at least one of A or B .
Symbolically:
A B = { x

U | x

A or x

B }
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Operations on Sets
•
Let A and B be subsets of a universal set U .
2. The intersection of A and B denoted
A B , is the set of all elements that are common to both A or B .
Symbolically:
A B = { x

U | x

A and x

B }
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Operations on Sets
•
Let A and B be subsets of a universal set U .
3. The difference of B minus A (or relative complement of A in B ) denoted B – A , is the set of all elements that are in B but not A .
Symbolically:
B – A = { x

U | x

B and x

A }
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Operations on Sets
•
Let A and B be subsets of a universal set U .
4. The complement of A denoted A c , is the set of all elements in U that are not A .
Symbolically:
A c = { x

U | x

A }
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Example – pg. 350 # 11
•
Let the universal set be the set R of all real numbers and let
A = { x

R | 0 < x

2}, B = { x

R | 1
 x < 4}, and
C = { x

R | 3
 x < 9}. Find each of the following: a. d. A C g. A c B c j. ( A B ) c
Erickson
A B b. A B e. A C h. A c B c c. A c f. B c i. ( A B ) c
6.1 Set Theory - Definitions and the
Element Method of Proof
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Unions and Intersections of an
Indexed Collection of Sets
Given sets A
0
, A
1
, A
2
, … that are subsets of a universal set U and given a nonnegative integer n , n
A i
  
|

A i
for at least one i

0,1, 2,..., n
 i

0
 i

0
A i
  
|

A i
for at least one nonnegative integer i
 n
A i
  
|

A i
for all i

0,1, 2,..., n
 i

0
 i

0
A i
  
|

A i
for all nonnegative integers i

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6.1 Set Theory - Definitions and the
Element Method of Proof
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Definitions
•
Empty Set
A set with no elements is called the empty set
(or null set ) and denoted by the symbol .
•
Disjoint
Two sets are called disjoint iff they have no elements in common. Symbolically:
A and B are disjoint A B =
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Definitions
•
Mutually Disjoint
Sets A
1
, A
2
, A
3
, … are mutually disjoint (or pairwise disjoint or nonoverlapping ) iff no two sets A i and A j with distinct subscripts have any elements in common. More precisely, for all i , j
= 1, 2, 3, …
A i
A j
= whenever i j .
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Example – pg. 305 # 23
V i

 x

| i
1 x i
1
 
 
1 1
Let for all i i 

 positive integers i .
4 a. i

1
V i
4

? b. i

1
V i

?
V V V
1 2 3
,... mutually disjoint? Explain.
n d. i

1
V i
 f. i

1
V i n

? e. i

1
V i


? g. i

1
V i


?
?
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6.1 Set Theory - Definitions and the
Element Method of Proof
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Definition
•
Partition
A finite or infinite collection of nonempty sets
{ A
1
, A
2
, A
3
, …} is a partition of a set A iff,
1.
A is the union of all the A i
2. The sets A
1
, A
2
, A
3
, …are mutually disjoint.
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6.1 Set Theory - Definitions and the
Element Method of Proof
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Example – pg. 351 # 27 b. Is
 
, ,
 
, ,
   
a partition of

, , , , , , ,

?
c. Is
         
a partition of

Erickson
       
a partition of

6.1 Set Theory - Definitions and the
Element Method of Proof
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Definition
•
Power Set
Given a set A , the power set of A is denoted
(A) , is the set of all subsets of A .
Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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Example – pg. 351 # 31
•
Suppose A = {1, 2} and B = {2, 3}. Find each of the following: a.

A B

b.
 c.

A B

d.

A B

Erickson
6.1 Set Theory - Definitions and the
Element Method of Proof
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