1. Discrete Mathematics
Set

2. Sets Set = a collection of distinct unordered objects
Members of a set are called elements
How to determine a set
Listing:
Example: A = {1,3,5,7}
Description
Example: B = {x | x = 2k + 1, 0 < k < 3}

3. Finite and infinite sets
Examples:
A = {1, 2, 3, 4}
B = {x | x is an integer, 1 < x < 4}
Infinite sets
Z = {integers} = {…, -3, -2, -1, 0, 1, 2, 3,…}
S={x| x is a real number and 1 < x < 4} = [0, 4]

4. Some important sets The empty set  has no elements.
Also called null set or void set.
Universal set: the set of all elements about which we make assertions.
Examples:
U = {all natural numbers}
U = {all real numbers}
U = {x| x is a natural number and 1< x<10}

5. Cardinality
Cardinality of a set A (in symbols |A|) is the number of elements in A
Examples:
If A = {1, 2, 3} then |A| = 3
If B = {x | x is a natural number and 1< x< 9}
then |B| = 9
Infinite cardinality
Countable (e.g., natural numbers, integers)
Uncountable (e.g., real numbers)

6. Subsets
X is a subset of Y if every element of X is also contained in Y
(in symbols X  Y)
Equality: X = Y if X  Y and Y  X
X is a proper subset of Y if X  Y but Y  X
Observation:  is a subset of every set

7. Power set
The power set of X is the set of all subsets of X, in symbols P(X),
i.e. P(X)= {A | A  X}
Example: if X = {1, 2, 3},
then P(X) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
If |X| = n, then |P(X)| = 2n.

8. Set operations: Union and Intersection
Given two sets X and Y
The union of X and Y is defined as the set
X  Y = { x | x  X or x  Y}
The intersection of X and Y is defined as the set
X  Y = { x | x  X and x  Y}
Two sets X and Y are disjoint if X  Y = 

9. Complement and Difference
The difference of two sets
X – Y = { x | x  X and x  Y}
The difference is also called the relative complement of Y in X
Symmetric difference
X Δ Y = (X – Y)  (Y – X)
The complement of a set A contained in a universal set U is the set Ac = U – A
In symbols Ac = U - A

10. Venn diagrams A Venn diagram provides a graphic view of sets
Set union, intersection, difference, symmetric difference and complements can be identified

11. Properties of set operations (1)
Theorem : Let U be a universal set, and A, B and C subsets of U. The following properties hold:
a) Associativity: (A  B)  C = A  (B  C)
(A  B)  C = A  (B C)
b) Commutativity: A  B = B  A
A  B = B  A

12. Properties of set operations (2)
c) Distributive laws:
A(BC) = (AB)(AC)
A(BC) = (AB)(AC)
d) Identity laws:
AU=A A = A
e) Complement laws:
AAc = U AAc = 

13. Properties of set operations (3)
f) Idempotent laws:
AA = A AA = A
g) Bound laws:
AU = U A = 
h) Absorption laws:
A(AB) = A A(AB) = A

14. Properties of set operations (4)
i) Involution law: (Ac)c = A
j) 0/1 laws: c = U Uc = 
k) De Morgan’s laws for sets:
(AB)c = AcBc
(AB)c = AcBc