Beginning Algebra, 11th Edition

598 APPENDIX A Sets

Distinguishing between Finite and Infinite Sets

List the elements of each set if possible. Decide whether each set is finite or infinite.

(a)The set of all integers

One way to list the elements is. The set is infinite.

(b)The set of all natural numbers between 0 and 5

The set is finite.

(c) The set of all irrational numbers

This is an infinite set whose elements cannot be listed. NOW TRY

Two sets are equal if they have exactly the same elements. Thus, the set of natural

numbers and the set of positive integers are equal sets. Also, the sets

and are equal.

The order of the elements does not make a difference.

OBJECTIVE 3 Decide whether a given set is a subset of another set.If all

elements of a set Aare also elements of another set B, then we say that Ais a subset

of B, written A 8 B.We use the symbol A 8 >Bto mean that Ais not a subset of B.

5 1, 2, 4, 7 6 5 4, 2, 7, 1 6

5 1, 2, 3, 4 6

5 Á, -2, -1, 0, 1, 2,Á 6

NOW TRY EXAMPLE 2

EXERCISE 2
List the elements of each set
if possible. Decide whether
each set is finite or infinite.

(a)The set of negative
integers

(b)The set of even natural
numbers between 11
and 19

Using Subset Notation

Let and Then

and

but and NOW TRY

The empty set is defined to be a subset of any set. Thus, the set has

four subsets:

and.

How many subsets does have? There is one subset with three ele-

ments: There are three subsets with two elements:

and

There are three subsets with one element:

and

There is one subset with no elements:. Thus, set Nhas eight subsets.

The following generalization can be made and proved in more advanced courses.

0

5 a 6 , 5 b 6 , 5 c 6.

5 a, b 6 , 5 a, c 6 , 5 b, c 6.

5 a, b, c 6.

N= 5 a, b, c 6

5 a, b 6 , 5 a 6 , 5 b 6 , 0

M= 5 a, b 6

AhB, AhC, BhC.

B 8 A, C 8 A, C 8 B,

A= 5 1, 2, 3, 4 6 ,B= 5 1, 4 6 , C= 516.

NOW TRY EXAMPLE 3

EXERCISE 3
Let

, and

Tell whether each statement is
trueor false.

(a) (b)

(c)ChA

B 8 A C 8 B

C= 5 1, 9, 11 6.

B= 5 1, 5, 7, 9 6

A= 5 1, 3, 5, 7, 9, 11 6 ,

NOW TRY ANSWERS

(a)
infinite
(b) 5 12, 14, 16, 18 6 ; finite

5 - 1, -2, -3, -4,Á 6 ;

(a)true (b)false (c)false

Number of Subsets of a Set

A set with nelements has 2 nsubsets.

To illustrate the relationships between sets, Venn diagramsare often used.

A rectangle represents the universal set, U. The sets under discussion are represented

by regions within the rectangle. The Venn diagram in FIGURE 1on the next page shows

that B 8 A.

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