Chapter 5 Section 3
Venn Diagram and Counting
Exercise 13 (page 222)
• Given:
n(U) = 20
n(S) = 12
n(T) = 14
n(S ∩ T ) = 18
• Problem, none of the values above
corresponds to any basic of the regions
Exercise 13 Solution
• Use the inclusion-exclusion principle
n( S U T ) = n( S ) + n( T ) – n( S ∩ T )
18
= 12 + 14 – n( S ∩ T )
n( S ∩ T ) = 8
Exercise 13 Venn Diagram
U
S
4
T
8
6
2
Basic region IV = 20 – ( 4 + 8 + 6 ) = 2
Exercise 16 (page 222)
• Given:
n(S) = 9
n(T) = 11
n(S ∩ T ) = 5
n(S´ ) = 13
**
• We can fill in all the basic regions except
for basic region I.
Exercise 16 Solution
• Use the following fact:
n( S ) + n( S ´ ) = n( U )
9
+ 13 = n( U )
n( U ) = 22
• Recall that all 4 basic regions must add up to n( U ).
• Thus:
Basic region IV = 22 – ( 4 + 5 + 6 ) = 7
With this information we then can fill in the Venn Diagram.
Exercise 16 Venn Diagram
U
S
4
7
T
5
6
Exercise 23 (page 222)
• First we need to define our sets:
• Let: S = { Students who like rock music }
T = { Students who like hip-hop music }
“Survey of 70 …students”
“35 students like rock music”
“15 students like hip-hop”
“5 liked both”
n(U) = 70
n(S) = 35
n(T) = 15
n(S ∩ T) = 5 **
Since n(S ∩ T) is basic region I in the two set Venn Diagram,
we can start filling in the Venn Diagram.
( ** means that this is a basic region in the Venn Diagram)
Exercise 23 Venn Diagram
U
S
30
T
5
10
25
Basic region IV = 70 – ( 30 + 5 + 10 ) = 25
Exercise 31 (page 223)
• First we need to define our sets:
• Let: U = { Students in Finite Math class }
M = { Male students in Finite Math class }
B = { Business students in Finite Math
class }
F = { First year students in Finite Math
class }
Exercise 31 Given
“35 students in class”
“22 are male students”
“19 are business students”
“27 are first-year students”
“17 are male first-year”
“15 are first-year business
“14 are male business”
“11 are male first-year business”
n(U) = 35
n(M) = 22
n(B) = 19
n(F) = 27
n( M ∩ F ) = 17
n( B ∩ F ) = 15
n( M ∩ B ) = 14
n( M ∩ B ∩ F ) = 11 **
( ** means that this is a basic region in the Venn Diagram)
Exercise 31 Venn Diagram
M
3
2
1
11
4
6
2
B
F
6
Basic Region VIII = 35 – ( 2 + 3 + 1 + 11 + 6 + 4 + 6 ) = 2
U