Chapter 2 Section 2 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 2 - Slide 2

Chapter 2

Sets

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 2 - Slide 3

WHAT YOU WILL LEARN• Methods to indicate sets, equal sets, and

equivalent sets• Subsets and proper subsets• Venn diagrams• Set operations such as complement,

intersection, union, difference and Cartesian product

• Equality of sets• Application of sets• Infinite sets

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 2 - Slide 4

Section 2

Subsets

Chapter 2 Section 2 - Slide 5Copyright © 2009 Pearson Education, Inc.

Subsets

A set is a subset of a given set if and only if all elements of the subset are also elements of the given set.

Symbol:

To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B. The symbol for “not a subset of” is .

Chapter 2 Section 2 - Slide 6Copyright © 2009 Pearson Education, Inc.

Example: Determine whether set A is a subset of set B.

A = { 3, 5, 6, 8 }B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Solution: All of the elements of set A are contained in set

B, so

Determining Subsets

A B .

Chapter 2 Section 2 - Slide 7Copyright © 2009 Pearson Education, Inc.

Some Facts

The empty set is a subset of every set, including itself.

Every set is a subset of itself.

Chapter 2 Section 2 - Slide 8Copyright © 2009 Pearson Education, Inc.

Proper Subset

All subsets are proper subsets except the subset containing all of the given elements, that is, the given set must contain one element not in the subset (the two sets cannot be equal).

Symbol:

Chapter 2 Section 2 - Slide 9Copyright © 2009 Pearson Education, Inc.

Determining Proper Subsets

Example: Determine whether set A is a proper subset of

set B.A = { dog, cat }B = { dog, cat, bird, fish }

Solution: All the elements of set A are contained in set B,

and sets A and B are not equal, therefore A B.

Chapter 2 Section 2 - Slide 10Copyright © 2009 Pearson Education, Inc.

Determining Proper Subsets (continued)

Example: Determine whether set A is a proper subset of

set B.A = { dog, bird, fish, cat }B = { dog, cat, bird, fish }

Solution: All the elements of set A are contained in set B,

but sets A and B are equal, therefore A B.

Chapter 2 Section 2 - Slide 11Copyright © 2009 Pearson Education, Inc.

Number of Distinct Subsets

The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.

Example: Determine the number of distinct subsets

for the given set { t , a , p , e }. List all the distinct subsets for the given set:

{ t , a , p , e }.

Chapter 2 Section 2 - Slide 12Copyright © 2009 Pearson Education, Inc.

Solution: Since there are 4 elements in the given set,

the number of distinct subsets is 24 = 2 • 2 • 2 • 2 = 16 subsets. {t,a,p,e},

{t,a,p}, {t,a,e}, {t,p,e}, {a,p,e}, {t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e},{t}, {a}, {p}, {e}, { }

Number of Distinct Subsets continued