Download set theroy_venn diagram_1999-2008

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.
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G
T
11
6
1
3
4
4
8
S
(a)
(b)
How many people
(i)
only play golf?
(ii)
play both tennis and golf?
(iii)
do not play golf?
Shade the part of the Venn diagram that represents the set G  S.
Working:
Answers:
(a) (i) ……………………………………..
(ii) …………………………………….
(iii) …………………………………….
(Total 4 marks)
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2.
In each of the Venn diagrams, shade the region indicated.
(a)
(b)
(c)
AB
A
B
A
B
A
B
The complement of (A  B)
The complement of (A  B)
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(d)
A  (B  C)
A
B
C
Working:
(Total 4 marks)
3.
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)
List the elements of
(i)
A;
(ii)
B.
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(b)
Place the elements of A and B in the appropriate region in the Venn diagram below.
U
A
B
Working:
Answers:
(a) (i) ………………………………………
(ii) ............................................................
(Total 4 marks)
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4.
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.
Answer:
(Total 4 marks)
5.
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)
Draw a Venn diagram to illustrate the above information.
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(b)
Calculate how many students
(i)
only watch TV;
(ii)
only go to the cinema.
Diagram:
Working:
Answers:
(b) (i) ……………………………………..
(ii) ……………………………………..
(Total 8 marks)
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6.
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
7
3
9
8
A
4
1
6
C
List the elements in each of the following sets.
(a)
AB
(b)
ABC
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(c)
(A’  C)  B
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
7.
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)
AC=
(b)
CB=C
(c)
C  (A  B)
(d)
A  C
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Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(d) ...................................................................
(Total 8 marks)
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8.
Let U = {–4, –
2
, 1, , 13, 26.7, 69, 1033}.
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A is the set of all the integers in U.
B is the set of all the rational numbers in U.
(a)
List all the prime numbers contained in U.
(b)
List all the members of A.
(c)
List all the members of B.
(d)
List all the members of the set A  B.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(d) ..................................................................
(Total 8 marks)
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9.
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
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(d)
A  C
U
B
A
C
(Total 8 marks)
10.
Given
the set of integers,
the set of rational numbers,
(a)
Write down an element that belongs to

.
(b)
Write down an element that belongs to

.
(c)
Write down an element that belongs to
(d)
Use a Venn diagram to represent the sets
the set of real numbers.
.
,
and
.
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) .....................................................
(Total 6 marks)
11.
12
f
4
A
5
1
–1
2
–2
–3
3
0
1
4
9
16
B
0
The diagram shows a function f, mapping members of set A to members of set B.
(a)
(i)
Using set notation, write down all members of the domain of f.
(ii)
Using set notation, write down all members of the range of f.
(iii)
Write down the equation of the function f.
The equation of a function g is g(x) = x2 +1. The domain of g is
(b)
.
Write down the range of g.
(Total 6 marks)
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12.
The Venn diagram below shows the universal set of real numbers
subsets:
and some of its important
: the rational numbers,
: the integers,
: the natural numbers.
Write the following numbers in the correct position in the diagram.
–1, 1, ,
7
, 3.333 3 ,
16
3.
Working:
(Total 6 marks)
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13.
At a certain school there are 90 students studying for their IB diploma. They are required to
study at least one of the subjects: Physics, Biology or Chemistry.
50 students are studying Physics,
60 students are studying Biology,
55 students are studying Chemistry,
30 students are studying both Physics and Biology,
10 students are studying both Biology and Chemistry but not Physics,
20 students are studying all three subjects.
Let x represent the number of students who study both Physics and Chemistry but not Biology.
Then 25–x is the number who study Chemistry only.
The figure below shows some of this information and can be used for working.
U with n(U)  90
Physics
x
Chemistry
25–x
20
10
Biology
20
(a)
Express the number of students who study Physics only, in terms of x.
(b)
Find x.
(c)
Determine the number of students studying at least two of the subjects.
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Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(Total 6 marks)
14.
B and C are subsets of a universal set U such that
U = {x : x 
(a)
, 0  x  10}, B = {prime numbers 10}, C = {x : x 
, 1  x  6}.
List the members of sets
(i)
B
(ii)
CB
(iii)
B  C′
Consider the propositions:
p: x is a prime number less than 10.
q: x is a positive integer between 1 and 7.
(b)
Write down, in words, the contrapositive of the statement, “If x is a prime number less
than 10, then x is a positive integer between 1 and 7.”
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Working:
Answers:
(a)
(i)..........................................
(ii).........................................
(iii)........................................
(b) ...................................................
...................................................
...................................................
...................................................
(Total 6 marks)
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15.
The figure below shows the set U of all functions of x.
A is the set of all sine and cosine functions of x (measured in degrees).
B is the set of all functions of x with period 120.
C is the subset of A, containing functions whose amplitude is 3.
U
A
C
B
Write the following functions in the correct place in the Venn diagram.
(You may use the numbers (i) to (vi) if you wish, to achieve a better fit.)
(i)
sin (x);
(ii)
x2;
(iii)
3 cos (x);
(iv)
2 sin (3x);
(v)
3 cos (3x);
(vi)
−3 sin (2x).
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Working:
(Total 6 marks)
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16.
The universal set U is the set of integers from 1 to 20 inclusive.
A and B are subsets of U where:
A is the set of even numbers between 7 and 17.
B is the set of multiples of 3.
List the elements of the following sets:
(a)
A;
(1)
(b)
B;
(1)
(c)
A  B;
(2)
(d)
A  B′.
(2)
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) .....................................................
(Total 6 marks)
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17.
A school offers three activities, basketball (B), choir (C) and drama (D). Every student must
participate in at least one activity.
16 students play basketball only.
18 students play basketball and sing in the choir but do not do drama.
34 students play basketball and do drama but do not sing in the choir.
27 students are in the choir and do drama but do not play basketball.
(a)
Enter the above information on the Venn diagram below.
(2)
U
B
C
x
D
99 of the students play basketball, 88 sing in the choir and 110 do drama.
(b)
Calculate the number of students x participating in all three activities.
(1)
(c)
Calculate the total number of students in the school.
(3)
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Working:
Answers:
(b) ...................................................
(c) ...................................................
(Total 6 marks)
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18.
(a)
Shade (A  B)  C′ on the diagram below.
U
B
A
C
(2)
(b)
In the Venn diagram below, the number of elements in each region is given.
Find n ((P  Q)  R).
U
P
2
3
1
Q
6
4
5
9
R
(2)
(c)
U is the set of positive integers,
E is the set of even numbers.
M is the set of multiples of 3.
+
.
(i)
List the first six elements of the set M.
(ii)
List the first six elements of the set E′  M.
(2)
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Working:
Answers:
(b) ..................................................
(c)
(i)..........................................
(ii).........................................
(Total 6 marks)
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