1. Discrete Mathematics CS 2610
January 27, 2009

2. Agenda Set Theory Set Builder Notation Universal Set
Power Set and Cardinality
Set Operations
Set Identities
Cartesian Product

3. Sets A set is an unordered collection of objects. Examples:
{ 1, 6, 7, 2, 9 }
{ a, d, e, 1, 2, 3}
Order and repetition don’t matter
= {6, 7, 1, 2, 9}
= {a, a, d, d, e, e, 1, 2, 3}
The empty set, or the set containing no elements.
 = {}
Note:   {}
Singleton is a set S that contains exactly one element

4. Universal Set
Universal Set is the set containing all the objects under consideration.
It is denoted by U

5. Set Builder Notation concise definition of a set
Set Builder – characterize the elements in a set by stating the properties that the elements must have to belong to the set.
{ x | P (x) }
reads x that satisfy P(x), x such that P(x)
x belongs to a universal set U.
concise definition of a set
Examples:
P = { x | x is prime number} U : Z+
M = { x | x is a mammal} U: All animals
Q+ = { x  R | x = p/q, for some positive integers p, q }

6. Elements of sets x  S means “x is an element of set S”
x  S means “x is not an element of set S
Example:
3  S reads:
“3 is an element of the set S ”.
Which of the following is true:
3  R
-3  N

7. Subsets A  B means “A is a subset of B” or, “B contains A”
“every element of A is also in B”
or, x ((x  A)  (x  B))
A  B means “A is a subset of B”
B  A means “B is a superset of A”

8. Subsets A  B means “A is a subset of B” For Every Set S,
i)   S, the empty set is a subset of every set
ii) S  S, every set is a subset of itself

9. Proper Subsets iff, A  B and A  B
A subset A of B is said to be a proper subset if A is not equal to B.
iff, A  B and A  B
iff, A  B and there is an x  B but x  A.
x ((x  A)  (x  B))  x ((x  B)  (x  A))
This is sometimes written A  B.

10. Set Equality
A = B if and only if A and B have exactly the same elements.
iff, A  B and B  A
iff, A  B and A  B
iff, x ((x  A)  (x  B)).
To show equality of sets A and B, must prove both:
A  B
B  A

11. Set Cardinality
The cardinality of a set is the number of distinct elements in the set. |S | denotes the cardinality of S.
S = {1,2,3}
|S| = 3
S = {5,5,5,5,5,5}
|S| = 1
S = 
|S| = 0
S = { , {}, {,{}} }
|S| = 3
A set S is said to be finite if its cardinality is a nonnegative integer. Otherwise, S is said to be infinite.
Given N = {0,1,2,3,…}, |N| is infinite (natural nos.)

12. Power Sets The power set of S is the set of all subsets of S.
P(S) = { x | x  S }
If S = {a}, P(S) = ?
{, {a}}
If S = {a,b}, P(S) = ?
{, {a}, {b}, {a, b}}
If S = , P(S)= ?
{}
Fact: if S is finite, |P(S)| = 2|S|.

13. n-Tuples An ordered n-tuple, n  Z+, is an ordered list
(a1, a2, …, an).
Its first element is a1.
Its second element is a2, etc.
Enclosed between parentheses (list not set).
Order and length matters:
(1, 2)  (2, 1)  (2, 1, 1).

14. Cartesian Product The Cartesian Product of two sets A and B is:
A x B = { (a, b) | a  A  b  B}
Example:
A= {a, b}, B= {1, 2}
A  B = {(a,1), (a,2), (b,1), (b,2)}
B  A = {(1,a), (1,b), (2,a), (2,b)}
Not commutative!
In general,
A1 x A2 x … x An = {(a1, a2,…, an) | a1  A1, a2  A2, …, an  An}
|A1 x A2 x … x An| = |A1| x |A2| x … x |An|

15. Union Operator The union of two sets A and B is:
A  B = { x | x  A v x  B }
Example:
A = {1,2,3}, B = {1,6}
A  B = {1,2,3,6}

16. Intersection Operator
The intersection of two sets A and B is:
A  B = { x | x  A  x  B}
Example:
A = {1,2,3}, B = {1,6}
A  B = {1}
Two sets A, B are called disjoint iff their intersection is empty.
A  B = 
A = {1,2,3}, B = {9,10}, C = {2, 9}
A and B are disjoint sets, but A and C are not

17. Set Theory : Inclusion/Exclusion
What is the cardinality of A  B ?
twice A B
AB
Once
|AB| = |A| + |B| - |A  B|

18. Set Complement The complement of a set A is: A = { x | x  A} Example:
U = N
A = {xN | x is odd }
A = {xN | x is even }
x  A  x  A
 = U
U = 

19. Set Difference The set difference, A - B, is:
A - B = { x | x  A  x  B }
Example:
A = {2,3,4,5 }, B = {3,4,7,9 }
A- B = {2, 5}
B – A = {7,9}
It is not commutative!!