c
Math 166, Fall 2015, Robert
Williams
1.2- The Number of Elements in a Set
If a set A has finitely many elements, we denote the number of elements in A
by n(A).
If A and B are disjoint, then n(A ∪ B) = n(A) + n(B).
For any finite sets A and B, n(A ∪ B) =
A
a
b
B
c
Example 1 Let A = {x, 6, z} and B = {x|x is a day in January 2015}
• n(A) =
• n(B) =
• n(∅) =
1
c
Math 166, Fall 2015, Robert
Williams
Example 2 The manager of a daycare decides to start a healthy snack program
by giving the children broccoli and carrots. When asking the children at the
daycare whether they like each snack, only 5 children said that they like carrots
and broccoli, while 74 children said they like carrots or broccoli, and 40 children
said they do not like broccoli. Find the number of children that only like carrots
and the number of children that only like broccoli.
A
B
b
a
c
Example 3 In order to determine how much to spend on a section of their
campaign, the Republican and Democrat canididates for an upcoming election
polled 2000 voters in a contested district. Of the voters polled, 1184 said they
would consider voting for the democrat, 1215 said they would consider voting
for the republican, and 213 said they would not vote for either candidate. How
many people could still be presuaded to vote for either candidate?
D
a
b
R
c
d
2
c
Math 166, Fall 2015, Robert
Williams
Example 4 As Blue Bell restarts ice cream production, they send vanilla, strawberry, and chocolate to a grocery store to test how many people want to buy their
products. The store’s records were incomplete, however, and they only got the
following information:
112 people bought vanilla
15 people bought all three
42 bought chocolate and strawberry
38 bought vanilla and strawberry but not chocolate
99 people bought two or more flavors
42 people bought only chocolate
106 bought chocolate or strawberry
101 did not buy strawberry
How many people bought only vanilla? Only chocolate?
How many people purchased ice cream?
V
C
b
a
c
e
d
f
h
g
S
3
c
Math 166, Fall 2015, Robert
Williams
Example 5 A travel agency is preparing promotional material for vacations to
France, Germany, and Italy. They decide to survey 150 customers to gauge how
popular each of their packages are. They learn the following:
47 customers have purchased a package for Germany, but not for Italy
65 customers have purchased a package for Germany or Italy, but not for France
66 customers have purchased a package for Italy
46 customers have purchased exactly two of the packages
42 customers have not purchased the packages for France or Germany
49 customers have not purchased the packages for France or Italy
17 customers have not purchased any of the three packages
How many people have only purchased a package for one of the destinations?
How many people have purchased all three?
F
G
b
a
c
e
d
f
h
g
I
4