1. Discrete Mathematics R. Johnsonbaugh
SETS

2. 2.1 Sets Set = a collection of distinct unordered objects
Members of a set are called elements
How to describe a set
Listing its elements
Example: A = {1,3,5,7} = {7, 5, 3, 1, 3}
Listing a property needed for membership
Example: B = {x | x = 2k + 1, 0 < k < 30}

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,…}
B={x| x is a positive, even integer}

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:
Z = {all Integers} i.e. -3, 0, 2, 145
R = {all real numbers} i.e. -3, , 0, 4/15, √2, π
Q = {rational numbers} i.e. -1/3, 0, 24/15

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,
i.e., X = Y whenever x  X, then x  Y,
and whenever x  X, then x  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}}
Theorem 2.1.4: If |X| = n, then |P(X)| = 2n.
|X| = 3 and |P(X)| = 23 = 8

8. Set operations: Produce a new set 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}
 : All elements belonging to X or Y or both
The intersection of X and Y is defined as the set
X  Y = { x | x  X and x  Y}
 : All elements belonging to both X and Y
Two sets X and Y are disjoint if X  Y = 

9. Complement and Difference
Complement: 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 set of all elements that belong to X or to Y but not both X and Y.
The complement of a set A contained in a universal set U is the set Ac = U – A = Ā

10. Example If X={1, 4, 7, 10}, Y={1, 2, 3, 4, 5} X  Y = X  Y = X – Y =

11. Example
If X={1, 4, 7, 10}, Y={1, 2, 3, 4, 5}
X  Y = {1, 2, 3, 4, 5, 7, 10}
X  Y = {1, 4}
X – Y = {7, 10}
Y – X = {2, 3, 5}

12. Venn diagrams A Venn diagram provides a graphic view of sets
Set union, intersection, difference, symmetric difference and complements can be easily and visually identified
VIDEO U C A B

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

14. Properties of set operations
Distributive laws:
A(BC) = (AB)(AC)
A(BC) = (AB)(AC)
Identity laws:
AU=A A = A
Complement laws:
AAc = U AAc = 

15. Properties of set operations
Idempotent laws:
AA = A AA = A
Bound laws:
AU = U A = 
Absorption laws:
A(AB) = A A(AB) = A

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

17. Addition Principle A.K.A The Inclusion-Exclusion Principle
If A and B are finite sets then,
| A  B | = |A| + |B| -
| A  B | U
A  B A B

18. Addition Principle for Disjoint Sets
| A  B  C | =
|A| + |B| + |C| - |A  B| - |B  C| - |A  C| + |A  B  C|
A = { a, b, c, d, e } B = { a, b, e, g, h } C = { b, d, e, g, h, k, m, n}
A company wants to hire 25 programmers to handle systems
Programming jobs and 40 programmers for applications programming.
Of those hired, ten will be expected to perform jobs of both types.
How many programmers must be hired?

19. One more example
A survey was taken on methods of commuter travel. Each respondent was asked to check BUS, TRAIN, or CAR as a major mode of traveling. More than one answer is allowed. The results are:
BUS TRAIN CAR 100
BUS and TRAIN BUS and CAR 15
TRAIN and CAR All three modes 5
How many people completed a survey form