Sets, Set Notation and Venn Diagrams - Worksheets, Questions and Revision

18 original exam-style questions with a full mark scheme - free printable PDF.

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GCSE · Number - Sets

N3 Sets, Set Notation and Venn Diagrams

AQA 8365 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
Which of the following correctly describes A intersect B, for two sets A and B?
  • A) All elements in A or B, or both
  • B) All elements that are in both A and B
  • C) All elements in A but not in B
  • D) All elements not in A
(Total for Question 1 is 1 mark)
2
Which of the following correctly describes A union B, for two sets A and B?
  • A) All elements in A and B
  • B) All elements in A or B, or both
  • C) All elements in A but not in B
  • D) The empty set
(Total for Question 2 is 1 mark)
3
The universal set is E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}. What is n(A)?
  • A) 2
  • B) 5
  • C) 8
  • D) 10
(Total for Question 3 is 1 mark)
4
The universal set is E = {1, 2, 3, ..., 20}. A is the set of even numbers in E. B is the set of multiples of 3 in E.
(a)List the elements of A intersect B.(2)
(b)Find n(A union B).(1)
(Total for Question 4 is 3 marks)
5
E = {x : x is an integer, 1 ≤ x ≤ 15}. C is the set of prime numbers in E. D is the set of factors of 12 in E.
(a)List the elements of C.(1)
(b)List the elements of C intersect D.(2)
(Total for Question 5 is 3 marks)
6
E = {1, 2, 3, ..., 10}. F = {2, 4, 6, 8, 10}. G = {1, 2, 3, 5, 7}.
(a)Write down the elements of F' (the complement of F).(1)
(b)List the elements of F' intersect G.(2)
(c)Find n(F union G).(1)
(Total for Question 6 is 4 marks)
7
E = {a, b, c, d, e}. P = {a, c, e}. Q = {b, d}.
(a)State, giving a reason, whether P and Q are disjoint (mutually exclusive) sets.(2)
(b)List all the subsets of Q.(2)
(Total for Question 7 is 4 marks)
8
40 students were surveyed about whether they play chess (C) or the guitar (G). 15 students play chess only, 8 students play both chess and the guitar, 12 students play the guitar only, and the rest play neither.
(a)Find the number of students who play neither chess nor the guitar.(2)
(b)Find n(C).(1)
(c)A student is chosen at random from the 40 students. Find the probability that the student plays the guitar but not chess.(1)
(Total for Question 8 is 4 marks)
9
In a survey of 200 people, 120 read newspaper X, 90 read newspaper Y, and 45 read both newspapers.
(a)Find the number of people who read newspaper X only.(1)
(b)Find the number of people who read neither newspaper.(2)
(c)A person is chosen at random. Find the probability that they read exactly one of the two newspapers.(1)
(Total for Question 9 is 4 marks)
10
50 people were asked whether they like tea (T) or coffee (Co). 20 people like tea only, x people like both tea and coffee, 3x people like coffee only, and 6 people like neither.
(a)Form an equation in x and solve it to find the value of x.(3)
(b)Find n(T union Co).(1)
(c)Find the probability that a randomly chosen person likes both tea and coffee.(1)
(Total for Question 10 is 5 marks)
11
E = {1, 2, 3, ..., 12}. A is the set of factors of 12. B is the set of even numbers in E. C is the set of multiples of 3 in E.
(a)List the elements of A intersect B intersect C (the numbers common to all three sets).(2)
(b)List the elements that are in both A and B, but not in C.(3)
(Total for Question 11 is 5 marks)
12
In a class of 45 students, every student studies at least one of Maths (M), Physics (P) and Chemistry (Ch). 5 students study all three subjects. 4 study Maths and Physics only. 3 study Maths and Chemistry only. 6 study Physics and Chemistry only. 10 study Maths only. 7 study Physics only. The rest study Chemistry only.
(a)Find the number of students who study Chemistry only.(2)
(b)Find n(M).(2)
(c)A student is chosen at random. Find the probability that the student studies exactly two of the three subjects.(2)
(Total for Question 12 is 6 marks)
13
A and B are two sets, both subsets of a universal set E. Explain why n(A union B) = n(A) + n(B) - n(A intersect B).
(Total for Question 13 is 3 marks)
14
A set X has n elements. Show that X has 2n subsets, illustrating your answer using a 3-element set.
(Total for Question 14 is 4 marks)
15
E = {x : x is an integer, -10 ≤ x ≤ 10}. A = {x : x2 ≤ 25, x is in E}. B = {x : x is a multiple of 3, x is in E}.
(a)List the elements of A.(2)
(b)List the elements of A intersect B.(2)
(c)Find n(A'), where A' is the complement of A within E.(1)
(Total for Question 15 is 5 marks)
16
The universal set is E = {1, 2, 3, 4, 5}. A = {1, 2}. B = {2, 3}. Determine whether (A union B)' = A' union B', justifying your answer.
(Total for Question 16 is 4 marks)
17
In a survey of 100 people, S, D and C are the sets of people who like swimming, diving and cycling respectively. 10 people like all three activities. The number who like exactly two of the three activities is 3 times the number who like all three. 25 people like swimming only, 20 like diving only, 15 like cycling only, and the remaining people like none of the three activities.
(a)Find the number of people who like exactly two of the three activities.(1)
(b)Find the number of people who like none of the three activities.(2)
(c)Given that a randomly chosen person from the survey likes at least two of the activities, find the probability that they like all three.(3)
(Total for Question 17 is 6 marks)
18
120 people were asked whether they read newspaper X or newspaper Y. The number who read only X is twice the number who read both. The number who read only Y is 10 more than the number who read both. 18 people read neither newspaper.
(a)Let b be the number of people who read both newspapers. Form an equation in b and solve it.(3)
(b)Find n(X), the number of people who read newspaper X.(2)
(c)Given that a randomly chosen person reads at least one of the two newspapers, find the probability that they read both.(2)
(Total for Question 18 is 7 marks)
Mark scheme · N3 Sets, Set Notation and Venn Diagrams

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18