Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

16 SET THEORY [CHAP. 1

(a) Find the number of people who read at least one of the three magazines.

(b) Fill in the correct number of people in each of the eight regions of the Venn diagram in Fig. 1-9(a)where

N,T, andFdenote the set of people who readNewsweek,Time, andFortune, respectively.

(c) Find the number of people who read exactly one magazine.

Fig. 1-9

(a) We want to findn(N∪T∪F). By Corollary 1.10 (Inclusion–Exclusion Principle),

n(N∪T∪F)=n(N )+n(T )+n(F )−n(N∩T)−n(N∩F)−n(T∩F)+n(N∩T∩F)

= 65 + 45 + 42 − 20 − 25 − 15 + 8 = 100

(b) The required Venn diagram in Fig. 1-9(b)is obtained as follows:

8 read all three magazines,

20 − 8 =12 readNewsweekandTimebut not all three magazines,

25 − 8 =17 readNewsweekandFortunebut not all three magazines,

15 − 8 =7 readTimeandFortunebut not all three magazines,

65 − 12 − 8 − 17 =28 read onlyNewsweek,

45 − 12 − 8 − 7 =18 read onlyTime,

42 − 17 − 8 − 7 =10 read onlyFortune,

120 − 100 =20 read no magazine at all.

(c)28+ 18 + 10 =56 read exactly one of the magazines.

1.16 Prove Theorem 1.9. SupposeAandBare finite sets. ThenA∪BandA∩Bare finite and

n(A∪B)=n(A)+n(B)−n(A∩B)

IfAandBare finite then, clearly,A∪BandA∩Bare finite.

Suppose we count the elements inAand then count the elements inB.

Then every element inA∩Bwould be counted twice, once inAand once inB. Thus

n(A∪B)=n(A)+n(B)−n(A∩B)