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

CHAP. 1] SET THEORY 15

1.13 Determine the validity of the following argument:

S 1 : All my friends are musicians.

S 2 : John is my friend.

S 3 : None of my neighbors are musicians.

S: John is not my neighbor.

The premisesS 1 andS 3 lead to the Venn diagram in Fig. 1-8(a).ByS 2 , John belongs to the set of friends which is

disjoint from the set of neighbors. ThusSis a valid conclusion and so the argument is valid.

Fig. 1-8

FINITE SETS AND THE COUNTING PRINCIPLE

1.14 Each student in LiberalArts at some college has a mathematics requirementAand a science requirementB.

A poll of 140 sophomore students shows that:

60 completedA, 45 completedB, 20 completed bothAandB.

Use a Venn diagram to find the number of students who have completed:

(a) At least one ofAandB; (b) exactly one ofAorB; (c) neitherAnorB.

Translating the above data into set notation yields:

n(A)=60,n(B)=45,n(A∩B)=20,n(U)= 140

Draw a Venn diagram of setsAandBas in Fig. 1-1(c). Then, as in Fig. 1-8(b), assign numbers to the four regions as

follows:

20 completed bothAandB,son(A∩B)=20.

60 − 20 =40 completedAbut notB,son(A\B)=40.

45 − 20 =25 completedBbut notA,son(B\A)=25.

140 − 20 − 40 − 25 =55 completed neitherAnorB.

By the Venn diagram:

(a)20+ 40 + 25 =85 completedAorB. Alternately, by the Inclusion–Exclusion Principle:

n(A∪B)=n(A)+n(B)−n(A∩B)= 60 + 45 − 20 = 85

(b)40+ 25 = 65 completed exactly one requirement. That is,n(A⊕B)=65.

(c) 55 completed neither requirement, i.e.n(AC∩BC)=n[(A∪B)C]= 140 − 85 =55.

1.15 In a survey of 120 people, it was found that:

65 readNewsweekmagazine, 20 read bothNewsweekandTime,

45 readTime, 25 read bothNewsweekandFortune,

42 readFortune, 15 read bothTimeandFortune,

8 read all three magazines.