APPENDIX

597

Sets

Listing the Elements of Sets

Represent each set by listing its elements.

(a)The set of states in the United States that border the Pacific Ocean is

California, Oregon, Washington, Hawaii, Alaska

(b)The set of all counting numbers less than

(c) The set of all counting numbers less than NOW TRY

In any discussion of sets, there is some set that includes all the elements under

consideration. This set is called the universal setfor that situation. For example, if

the discussion is about presidents of the United States, then the set of all presidents of

the United States is the universal set. The universal set is denoted U.

OBJECTIVE 2 Decide whether a set is finite or infinite.In Example 1,there

are five elements in the set in part (a) and five in part (b). If the number of elements

in a set is either 0 or a counting number, then the set is finite.By contrast, the set of

natural numbers is an infiniteset, because there is no final natural number. We can

list the elements of the set of natural numbers as

where the three dots indicate that the set continues indefinitely. Not all infinite sets

can be listed in this way. For example, there is no way to list the elements in the set of

all real numbers between 1 and 2.

N= 5 1, 2, 3, 4,Á 6 ,

0 = 0

6 = 5 1, 2, 3, 4, 5 6.

5 6.

EXAMPLE 1

OBJECTIVES

1 Learn the

vocabulary and

symbols used to

discuss sets.

2 Decide whether

a set is finite or

infinite.

3 Decide whether a

given set is a subset

of another set.

4 Find the

complement

of a set.

5 Find the union and

the intersection of

two sets.

OBJECTIVE 1 Learn the vocabulary and symbols used to discuss sets.A

setis a collection of objects. These objects are called the elementsof the set. A set is

represented by listing its elements between braces, .* The order in which the

elements of a set are listed is unimportant.

Capital letters are used to name sets. To state that 5 is an element of

write The statement means that 6 is not an element of S.

The set with no elements is called the empty set,or the null set.The symbol or

is used for the empty set. If we let Abe the set of all negative natural numbers,

then Ais the empty set.

A= 0 or A= 5 6

5 6

0

5 S. 6 S

S= 5 1, 2, 3, 4, 5 6 ,

5 6

*Some people refer to this convention as roster notation.

NOW TRY
EXERCISE 1
List the elements of the set
of odd natural numbers less
than 13.

CAUTION Do not make the common error of writing the empty set as 506.

NOW TRY ANSWER

1. 5 1, 3, 5, 7, 9, 11 6

A