7F: Numbers in Regions
7G: Problem Solving with Venn Diagrams
Objective: To solve problems by creating Venn diagrams.
Shade the regions that apply:
a) in A, but not B
b) in B, but not A
c) in both A and B d) neither in A nor in B e) either A or B
Examples:
1. If (3) means that there are 3 elements in the set A  B, how many elements are there in:
a) A
b) B’
c) A  B
d) A, but not B
e) B, but not A
f) neither A nor B
(7) (3) (11)
(4)
2. Given n(U) = 30, n(A) = 14, n(B) = 17 and n(A  B) = 6, find:
a) n(A  B)
b) n(A, but not B)
3. A squash club has 27 members. 19 have black hair, 14 have brown eyes and 11 have both black hair and
brown eyes. Place this information on a Venn diagram. Hence, find the number of members with:
a) black hair or brown eyes
b) black hair, but not brown eyes
4. A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from
4 m. How many dive from both platforms?
5. A city has three newspapers A, B, and C. Of the adult population,
1% read none of these newspapers, 36% read A, 40% read B, 52% read
C, 8% read A and B, 11% read B and C, 13% read A and C and 3%
read all three papers. What percentage of the adult population read:
a) newspaper A only
b) newspaper B or newspaper C
c) newspaper A or B but not C
Practice:
1. In a year group of 63 students, 22 study Biology, 26 study Chemistry and 25 study Physics. 18 study both
Physics and Chemistry, four study both Biology and Chemistry and three study both Physics and Biology.
One studies all three subjects. How many students study:
a) Biology only
b) Physics or Chemistry
c) none of Biology, Physics, or Chemistry d) Physics but not chemistry?
2. In a factory, 56 people work on the assembly line. 47 work day shifts and 29 work night
shifts. How many work both day shifts and night shifts?
3. Given n(U) = 50, n(S) = 30, n(R) = 25 and n(R  S) = 48, find:
a) n(R  S)
b) n(S, but not R)
4. There were 32 students available for the woodwind section of the school orchestra. 11 students could play
the flute, 15 could play the clarinet and 12 could play the saxophone. Two could play the flute and the
saxophone, two could play the flute and the clarinet and 6 could play the clarinet and the saxophone. One
student could play all three instruments. Find the number of students who could play:
a) woodwind instruments other than the flute, clarinet and saxophone
b) only the saxophone
c) the saxophone and clarinet but not the flute
d) only one of the clarinet, saxophone or flute.