Sets

1
Sets

• Section 2.1

2
Sets

• Set - a collection of unordered objects
• Element /member - an object in a set
• Two ways of describing a Set
• Specifying the elements explicitly
• V a, e, i, o, u
• Specifying the property/properties of the
  elements (Set Builder notation)
• P x x is a positive integer less than 10

3
Set membership and common sets

• Set membership
• a ? A a is an element of set A
• b ? A b is not an element of set A
• Some notations
• N Set of natural numbers
• Z Set of integers
• Z Set of positive integers
• Q Set of rational numbers
• R Set of real numbers

4
Set Definitions

• Equal sets
• Two sets are equal iff they have the same
  elements
• Order means nothing
• Listing an object more than once does not change
  the set

5
Set Definitions (Cont..)

• Empty set or Null set
• Set that has no elements
• Denoted by ? or
• Singleton
• Set that has one element
• Example A 2

6
Set Definitions (Cont..)

• The set A is a subset of B iff every element of A
  is also an element of B.
• A ? B
• The null set is a subset of every set.
• ? ? A
• Every set is a subset of itself.
• A ? A
• A is a proper subset of B, if A is a subset of B
  and A is not equal to B.
• A ? B

7
Set Definitions (Cont..)

• Cardinality
• The number of distinct elements in a set
• Denoted by S, for the set S

8
Power Set

• Power Set
• The set of all subsets of a set
• Denoted by P(S), for the set S
• If a set has n elements, then the power set has
  2n elements, i.e., P(S) 2S

9
Example

• Let L a, b, c, d
• What is the cardinality of L?
• How many elements does the power set of L have?
• What is the power set of L?

10
Venn Diagram

• Universal set U is represented by rectangle
• Circles or other geometrical figures inside the
  rectangle represent sets

11
Cartesian Product

• The Cartesian product of two sets A and B
  (denoted by A ? B) is the set of all ordered
  pairs (a,b) where a is an element of A and b is
  an element of B.
• A ? B (a,b) a?A ? b?B
• Example A x, y, z B 1, 2
• A ? B (x,1),(x,2),(y,1),(y,2),(z,1),(z
  ,2)
• B ? A (1,x),(1,y),(1,z),(2,x),(2,y),(2
  ,z)

12
Set Operations

• Section 2.2

13
Set Union

• Union of two sets A and B is denoted by A?B
• A?B contains elements that are either in A or in
  B or in both
• A?B x x?A ? x?B
• A 1,3,5, B 2,3,4

1,2,3,4,5

• A?B

14
Set Intersection

• Intersection of two sets A and B is denoted by
  A?B
• A?B contains elements that are in both A and B
• A?B x x?A ? x?B
• A 1,3,5, B 1,2,3

1,3

• A?B

15
Disjoint Sets

• Two sets are called disjoint if their
  intersection is the empty set.
• A 1,3,5, B 1,2,3, C 6,7,8

NO

• Are A and B disjoint?

YES

• Are A and C are disjoint?

16
Cardinality of union of sets

• Exercise
• How many elements does A U B have??
• A?B AB-A ? B

17
Set Difference

• Difference of two sets A and B is denoted by A?B
• A?B contains elements that are in A but not in B.
• A?B x x?A ? x?B
• A 1,3,5, B 1,2,3

5

• A?B

18
Complement of a Set

• Done with respect to a Universal set U

19
Set Identities
20
Set Identities (Cont ..)
21
Examples

• Use set builder notation to prove that

• Use set identities to prove that

22
More Exercises

• Describe the following sets using the set
• builder notation
• 1. The set of all positive integers
  between 1 and 99.
• 2.
• 3.
• 4.
• 5.
• Use set builder notation to prove
  .