IBMS I Sets and Venn diagrams practice
Be sure to show all work!
1.
The sports offered at a retirement village are Golf (G), Tennis (T) and Swimming
(S). The Venn diagram shows the numbers of people involved in each activity.
2
G
T
11
6
1
3
4
4
8
S
(a)
How many people
(i)
only play golf?
(ii)
play both tennis and golf?
(iii) do not play golf?
(b)
Shade the part of the Venn diagram that represents the set G  S.
(Total 4 marks)
2.
Let ξ = {x : 1 ≤ x < 17, x  }.
P , Q and R are the subsets of ξ such that
P = {multiples of four};
Q = (factors of 36};
R = {square numbers}.
(a)
List the elements of
(i)
ξ
(ii)
P  Q  R.
(2)
(b)
Describe in words the set P  Q.
(1)
1
(c)
(i)
Draw a Venn diagram to show the relationship between sets P, Q and
R.
(2)
(ii)
Write the elements of ξ in the appropriate places on the Venn diagram.
(3)
3.
In each of the Venn diagrams, shade the region indicated.
(a)
AB
A
(b)
B
The complement of (A  B)
A
(c)
(d)
B
The complement of (A  B)
A
B
A
B
A  (B  C)
C
(Total 4 marks)
4.
Let
ξ = {positive integers less than 15};
X= {multiples of 2};
Y = {multiples of 3}.
(a)
Show, in a Venn diagram, the relationship between the sets ξ, X and Y.
(1)
2
(b)
List the elements of:
(i)
XY
(1)
(ii)
X  Y.
(2)
(c)
Find the number of elements in the complement of (X  Y).
(2)
(Total 6 marks)
5.
A group of 30 children are surveyed to find out which of the three sports cricket
(C), basketball (B) or volleyball (V) they play. The results are as follows:
3 children do not play any of these sports
2 children play all three sports
6 play volleyball and basketball
3 play cricket and basketball
6 play cricket and volleyball
16 play basketball
12 play volleyball.
(a)
Draw a Venn diagram to illustrate the relationship between the three sports
played.
(1)
(b)
On your Venn diagram indicate the number of children that belong to each
region.
(3)
(c)
How many children play only cricket?
(2)
(Total 6 marks)
6.
In a club with 60 members, everyone attends either on Tuesday for Drama (D) or
on Thursday for Sports (S) or on both days for Drama and Sports.
One week it is found that 48 members attend for Drama and 44 members attend
for Sports and x members attend for both Drama and Sports.
(a)
(i)
Draw and label fully a Venn diagram to illustrate this information.
(3)
(ii)
Find the number of members who attend for both Drama and Sports.
(2)
(iii) Describe, in words, the set represented by (D  S)'.
(2)
3
7.
The universal set U is defined as the set of positive integers less than 10. The
subsets A and B are defined as:
A = {integers that are multiples of 3}
B = {integers that are factors of 30}
(a)
(b)
List the elements of
(i)
A;
(ii)
B.
Place the elements of A and B in the appropriate region in the Venn diagram
below.
U
A
B
(Total 4 marks)
8.
The sets A, B and C are subsets of U. They are defined as follows:
U = {positive integers less than 16}
A = {prime numbers}
B = {factors of 36}
(a)
C = {multiples of 4}
List the elements (if any) of the following:
(i)
A;
(ii)
B;
(iii) C;
(iv) A  B  C.
(4)
(b)
(i)
Draw a Venn diagram showing the relationship between the sets U, A,
B and C.
(ii)
Write the elements of sets U, A, B and C in the appropriate places on the
Venn diagram.
(4)
4
(c)
From the Venn diagram, list the elements of each of the following
(i)
A  (B  C);
(ii)
(A  B);
(iii) (A  B)  C.
(3)
9.
A committee U has three sub-committees: research R, finance F and purchasing P.
No member belongs to both finance and purchasing sub-committees. Some
members belong to both research and purchasing committees. All members of the
finance sub-committee also belong to the research sub-committee.
Draw a Venn diagram, showing the relationship between the sets U, R, F and P.
(Total 4 marks)
10.
100 students were asked which television channel (MTV, CNN or BBC) they had
watched the previous evening. The results are shown in the Venn diagram below.
U
MTV
CNN
35
19
23
6
5
2
3
7
BBC
From the information in the Venn diagram, write down the number of students
who watched
(a)
both MTV and BBC;
(b)
MTV or BBC;
(c)
CNN and BBC but not MTV;
(d)
MTV or CNN but not BBC.
(Total 4 marks)
5
11.
The Venn diagram below shows the number of students studying Science (S),
Mathematics (M) and History (H) out of a group of 20 college students. Some of
the students do not study any of these subjects, 8 study Science, 10 study
Mathematics and 9 study History.
U
1
S
M
4
A
2
1
3
3
(a)
H
(i)
How many students belong to the region labelled A?
(ii)
Describe in words the region labelled A.
(iii) How many students do not study any of the three subjects?
(5)
(b)
Draw a sketch of the Venn diagram above and shade the region which
represents S   H.
(1)
(c)
Calculate n(S  H).
(2)
12.
A poll was taken of the leisure time activities of 90 students.
60 students watch TV (T), 60 students read (R), 70 students go to the cinema (C).
26 students watch TV, read and go to the cinema.
20 students watch TV and go to the cinema only.
18 students read and go to the cinema only.
10 students read and watch TV only.
(a)
(b)
Draw a Venn diagram to illustrate the above information.
Calculate how many students
(i)
only watch TV;
(ii)
only go to the cinema.
(Total 8 marks)
6
13. In the Venn diagram below, A, B and C are subsets of a universal set U =
{1,2,3,4,6,7,8,9}.
U
B
2
List the elements in each of the following sets.
(a)
AB
(b)
ABC
(c)
(A’  C)  B
7
3
9
8
A
1
4
6
C
(Total 8 marks)
14.
The sets U, P, R and S are defined as follows:
U = {all quadrilaterals}
P = {all parallelograms}
R = {all rectangles}
S = {all squares}
(a)
Draw a Venn Diagram illustrating the relationships of the above sets.
(b)
Draw a separate Venn Diagram for each of the examples below.
Indicate by shading each of the following:
(i)
(P  S)
(ii)
(R  S)  P
(4)
(4)
(Total 8 marks)
15.
The following Venn Diagram shows the sets U, A, B and C.
U
B
A
C
State whether the following statements are true or false for the information
illustrated in the Venn Diagram.
(a)
(c)
A  C = 
C  (A  B)



(b)
(d)
CB=C
A  C
(Total 8 marks)
7
16.
On a particular day 100 children are asked to make a note of what they drank that
day.
They are given three choices: water (W), coffee (C) or fruit juice (F)
1 child drank only water.
6 children drank only coffee.
8 children drank only fruit juice.
5 children drank all three.
7 children drank water and coffee only.
53 children drank coffee and fruit juice only.
18 children drank water and fruit juice only.
17.
(a)
Represent the above information on a Venn Diagram.
(4)
(b)
How many children drank none of the above?
(2)
Shade the given region on the corresponding Venn Diagram.
(a)
AB
U
B
A
C
(b)
CB
U
B
A
C
(c)
(A  B C)
U
B
A
C
8
(d)
A  C
U
B
A
C
(Total 8 marks)
18.
A school jazz band contains three different musical instruments – saxophone (S),
clarinet (C) and drums (D). Students in the band are able to play one, two or three
different instruments.
In a class of 40 IB students, 25 belong to the jazz band. Out of these 25
3 can play all three instruments
5 can play the saxophone and clarinet only
5 can play at least the clarinet and drums
7 can play at least the saxophone and drums
16 can play the saxophone
12 can play the clarinet.
19.
(a)
Draw a Venn Diagram and clearly indicate the numbers in each region.
(5)
(b)
Show that the number of students who play the drums only is 5.
(2)
Given the set of integers,
numbers.
the set of rational numbers,
(a)
Write down an element that belongs to

.
(b)
Write down an element that belongs to

.
(c)
(d)
Write down an element that belongs to .
Use a Venn diagram to represent the sets ,
and
the set of real
.
(Total 6 marks)
(Total 6 marks)
20.
The following results were obtained from a survey concerning the reading habits
of students.
60 % read magazine P
50 % read magazine Q
50 % read magazine R
30 % read magazines P and Q
20 % read magazines Q and R
30 % read magazines P and R
9
10 % read all three magazines
(a)
Represent all of this information on a Venn diagram.
(4)
(b)
What percentage of students read exactly two magazines?
(1)
(c)
What percentage of students read at least two magazines?
(1)
(d)
What percentage of students do not read any of the magazines?
(1)
(Total 7 marks)
10