Beginning Algebra, 11th Edition

OBJECTIVE 4 Find the complement of a set.For every set A, there is a set

the complementof A, that contains all the elements of Uthat are not in A. The

shaded region in the Venn diagram in FIGURE 2represents A¿.

Aœ,

Appendix A Sets

FIGURE 1

U

B A

FIGURE 2

U

A′ is shaded.

A

A′

Determining Complements of a Set

Given and list

the elements of and.

and NOW TRY

OBJECTIVE 5 Find the union and the intersection of two sets. The union

of two sets Aand B, written is the set of all elements of Atogether with all

elements of B. Thus, for the sets in Example 4,

and

In FIGURE 3, the shaded region is the union of sets Aand B.

A ́B= 5 a, b, c, d, f, g 6 A ́C = 5 a, b, c, d, e 6.

A ́B,

A¿= 5 d, e, f, g 6 , B¿= 5 b, c, e 6 , C¿= 5 a, b, c, f, g 6.

A¿,B¿, C¿

U= 5 a, b, c, d, e, f, g 6 ,A= 5 a, b, c 6 ,B= 5 a, d, f, g 6 , C = 5 d, e 6 ,

EXAMPLE 4

Finding the Union of Two Sets

If and find

NOW TRY

The intersectionof two sets Aand B, written is the set of all elements that

belong to both Aand B. For example, if

and

then A ̈B= 5 José, Ellen 6.

B= 5 José, Patrick, Ellen, Sue 6 ,

A= 5 José, Ellen, Marge, Kevin 6

A ̈B,

M ́N = 5 1, 2, 3, 4, 5, 7 6

M= 5 2, 5, 7 6 N= 5 1, 2, 3, 4, 5 6 , M ́N.

EXAMPLE 5

FIGURE 3

U

A B

A ∪ B is shaded.

NOW TRY EXERCISE 4

Let

and.

List the elements in M¿.

M= 5 2, 10, 12, 14 6

U= 5 2, 4, 6, 8, 10,12, 14 6

NOW TRY ANSWERS

5 4, 6, 8 6

NOW TRY
EXERCISE 5
If and
N= 5 0, 3, 6, 9 6 ,find M ́N.

M= 5 1, 3, 5, 7, 9 6

5 0, 1, 3, 5, 6, 7, 9 6