Math 1313
Section 5.2
1
Section 5.2 - The Number of Elements in a Set
Let A be a set, then n(A) is the
Example 1: Let A = {1, 2, 3, …, 19, 20} and B = {q, s, t, v}.
Find:
a. n(A) =
b. n(B) =
c. n(Ø) =
Example 2: In a survey of 374 coffee drinkers it was found that 227 take
sugar, 245 take cream, and 163 take both sugar and cream with their coffee.
Define your sets first.
a. How many coffee drinkers take sugar or cream with their coffee?
b. How many take sugar only?
c. How many do not take sugar or cream?
d. How many take sugar or cream, but not both?
Math 1313
Section 5.2
2
Given two sets A and B:
1. If A and B are disjoint then n( A U B ) = n( A ) + n ( B ) .
2. If A and B are not disjoint then n( A U B ) = n( A ) + n( B ) − n( A I B )
Example 3: Let A and B be subsets of a universal set U. Given that n(B) = 9,
n(A I B) = 5, and n(A U B) =20, find n(A).
Example 4: Let A and B be subsets of a universal set U. Given that n(A) = 10,
n( A c ∩ B) = 12 , n( A I B ) = 4 , and n( A U B )c = 9 , find n( A U B c ) .
U
A
B
Math 1313
Section 5.2
3
Example 5: Suppose n(U) = 200,
∪
∩
= . Fine the following:
a.
=
,
=
, and
∪
U
A
B
b.
∩
Example 6: Of 50 employees of a store located in downtown Boston. 18 people
take the subway to work, 12 take the bus and 7 take the bus and the subway.
How many employees;
a. Take the subway or the bus?
b. Take the bus only to work?
c. Take either the bus or the subway to work?
d. Get to work by some other measures?
U
Math 1313
Section 5.2
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Example 7: Let n(U) = 77, n(A) = 45, n(B) = 40, n(C) = 41, n(A I B) = 24,
n(B I C) = 22, n(A I C) = 30, and n(A I B I C) = 16. Find the number in each of
the following sets.
U
A
B
C
a.
b.
c.
∪
∩
∪
∪
∩
Math 1313
Section 5.2
5
Example 8: 190 students enrolled in last summer classes. 6 had all three
classes. 35 had Math only, 60 had History only, and 32 had Speech only. 3 had
both History and Speech only, 20 had both Math and Speech only and 18 had
both Math and History only.
Define your sets first.
U
M
H
S
a. How many students surveyed enrolled Math?
b. How many students surveyed enrolled in none of these classes?
c. How many students surveyed enrolled at most one of these three subjects
mentioned?
d. How many of these students were enrolled in two of these subjects?
Math 1313
Section 5.2
6
Example 9: To help plan the numbers of meals to be prepared in a college
cafeteria, a survey was conducted and the following data was obtained:
130 students ate breakfast.
180 students ate lunch.
275 students ate dinner.
68 students ate breakfast and lunch.
112 students ate breakfast and dinner.
90 students ate lunch and dinner.
58 students at all three meals.
U
How many of the students:
a. Ate at least one meal in the cafeteria?
b. Ate exactly one meal in the cafeteria?
c. Ate only dinner in the cafeteria?
d. Ate exactly two meals in the cafeteria?