Section 6.2 The Number of Elements in a Finite Set

Counting Problems If a problem requires knowing the number of elements in a given set, then we call such a problem a Counting problem.

Number of Elements in A If A is a set, then n(A)isthenumberofelementsinthesetA.IfA is a finite set, then we can simply count the number of elements in A to find n(A). Note: If U is a universal set and A is a subset of U,thenn(Ac)=n(U) n(A) 1. Let the universal set U = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 . Find the following. { } (a) n(U)

(b) n(Ac), where A = x xisanevennumberfrom1to10 { | }

(c) n(B), where B = 1, 3, 9 { }

(d) n(?)

Addition Rule for Sets: Very Useful Formula If A and B are finite sets then

n(A B)=n(A)+n(B) n(A B) [ \

2. If n(B)=13,n(A B)=24,andn(A B)=6,findn(A). [ \ 3. In a survey of 272 people, a pet food manufacturer found that 69 owned a dog but not a cat, 28 owned a cat but not a dog, and 73 owned neither a dog or a cat.

(a) How many owned both a cat and a dog? (b) How many owned a dog?

Number of Subsets Suppose A is a set and that n(A)=m,wherem is any nonnegative integer. Then the number of subsets of A is 2m.Thenumberofproper subsets of A is 2m 1.

4. Let A and B be subsets of a universal set U and suppose n(U)=48,n(A)=13,n(B)=23,and n(A B) = 8. Compute: \ (a) n(Ac B) \ (b) n(Bc) (c) n(Ac Bc) \ (d) How many subsets does A have? (e) How many proper subsets does A have?

2 Fall 2017, Maya Johnson 5. Let A, B,andC be sets in a universal set U.Wearegivenn(U)=66,n(A)=32,n(B)=33, n(C)=33,n(A B)=16,n(A C)=10,n(B C)=18,n(A B Cc) = 9. Find the following \ \ \ \ \ values.

(a) n((A B C)c) [ [ (b) n(Ac Bc C) \ \

3 Fall 2017, Maya Johnson 6. Use the following information to determine the number of people in each region of the Venn Diagram. Agroupof295studentswereaskedwhichofthesesportstheyparticipatedinduringhighschool.

44 students participated in all of these sports.

87 students participated in basketball and track.

39 students participated in basketball and tennis but not track.

79 students participated in track but not tennis.

155 students participated in basketball.

142 students did not participate in tennis.

103 students participated in exactly one sport.

a =

b = Track Tennis c = a b c d = e d f e =

g f =

Basketball h g =

h =

- = - 32 - - 29 34 Fossett 44 43

4 Fall 2017, Maya Johnson