Sets and Venn Notation - Worksheets, Questions and Revision

12 original exam-style questions - 5 pages of questions with a full mark scheme - free printable PDF.

Download PDF Jump to mark scheme (page 6)Sets and Venn Notation revision guide

This topic is chapter 1 of IGCSE Maths Practice Book 1.

1.1 Sets and Venn Notation

EDEXCEL 4MA1 · Calculator allowed · about 65 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Using the sets E and M from Question 3, complete each statement with 'is a member of' or 'is not a member of'.

(i)9 ..... E(1)

(ii)9 ..... M(1)

(Total for Question 1 is 2 marks)

The universal set is xi = {1, 2, 3, ..., 20}. A is the set of multiples of 4 in xi. B is the set of multiples of 6 in xi.

(a)List the elements of A.(1)

(b)List the elements of B.(1)

(c)List the elements of A intersection B.(1)

(d)List the elements of A union B.(1)

(e)Find n((A union B)').(1)

(Total for Question 2 is 5 marks)

The universal set is xi = {1, 2, 3, ..., 12}. E = {even numbers in xi}. M = {multiples of 3 in xi}. Find the elements of E intersection M.

(Total for Question 3 is 2 marks)

The universal set is xi = {1, 2, 3, ..., 10}. A = {2, 4, 6, 8, 10}. B = {4, 8}.

(a)Is B a subset of A? Give a brief justification.(1)

(b)List all the subsets of B that contain exactly one element.(2)

(Total for Question 4 is 3 marks)

P = {x : x is a prime number, x < 20}. List the elements of P.

(Total for Question 5 is 2 marks)

R = {even numbers from 1 to 9} and S = {odd numbers from 1 to 9}. Find R intersection S, giving a reason for your answer.

(Total for Question 6 is 2 marks)

The Venn diagram shows two sets, A and B, inside the universal set xi. The region inside circle A but outside circle B is shaded. Which of the following correctly describes the shaded region?

Figure: A two-circle Venn diagram, circles A and B overlapping, inside a rectangle labelled xi. Only the part of circle A that does not overlap with circle B is shaded.

A) A intersection B
B) A union B
C) A intersection B'
D) A' intersection B

(Total for Question 7 is 1 mark)

Using the same two-circle Venn diagram as Question 7 (circles A and B inside universal set xi), write down the set notation for each region described below.

(a)The region where circles A and B overlap.(1)

(b)The region outside both circles.(1)

(c)The region inside circle B but not inside circle A.(1)

(Total for Question 8 is 3 marks)

Using the sets E and M from Question 3 (xi = {1, ..., 12}, E = {2,4,6,8,10,12}, M = {3,6,9,12}), find n(E union M).

(Total for Question 9 is 3 marks)

In a group of 60 students, 34 like Football (F), 28 like Netball (N), and 10 like neither sport.

Figure: A two-circle Venn diagram inside a rectangle labelled xi. The circles are labelled F and N and overlap. Numeric values are not pre-filled; this diagram is to be completed as part of solving the question.

(a)Find n(F union N).(2)

(b)Find n(F intersection N).(2)

(c)Find the number of students who like Football only (Football but not Netball).(1)

(d)A student is chosen at random from the group of 60. Find the probability that the student likes Netball but not Football. Give your answer as a fraction in its simplest form.(2)

(Total for Question 10 is 7 marks)

A and B are sets, with universal set xi. n(xi) = 50, n(A) = 22, n(B) = 19, and n(A union B) = 35. Find n(A intersection B).

(Total for Question 11 is 3 marks)

In a survey of 100 students, the number studying French (F), German (G) and Spanish (S) is shown in the Venn diagram: the region where all three subjects overlap contains 5 students; the region for F and G only (not S) contains x students; the region for F and S only (not G) contains 8 students; the region for G and S only (not F) contains 6 students; the region for F only contains 20 students; the region for G only contains 15 students; the region for S only contains 2x students; and 10 students, outside all three circles, study none of the three languages.

Figure: A three-circle Venn diagram (circles F, G and S, all overlapping in the classic three-set arrangement) inside a rectangle labelled xi. Region values: all three = 5, F and G only = x, F and S only = 8, G and S only = 6, F only = 20, G only = 15, S only = 2x, outside all three circles = 10.

(a)Form an equation in x, and solve it to find the value of x.(3)

(b)Find the number of students who study Spanish.(3)

(c)Find the number of students who study at least two of the three languages.(2)

(d)A student is selected at random from the 100 students. Find the probability that the student studies exactly one of the three languages.(2)

(Total for Question 12 is 10 marks)

Mark scheme · 1.1 Sets and Venn Notation

Question 1

(i) B1 9 is not a member of E
(i) Answer: 9 is not a member of E
(ii) B1 9 is a member of M
(ii) Answer: 9 is a member of M

Question 2

(a) B1 A = {4, 8, 12, 16, 20} cao
(a) Answer: A = {4, 8, 12, 16, 20}
(b) B1 B = {6, 12, 18} cao
(b) Answer: B = {6, 12, 18}
(c) B1 A intersection B = {12} cao, ft from parts (a) and (b)
(c) Answer: A intersection B = {12}
(d) B1 A union B = {4, 6, 8, 12, 16, 18, 20} cao, ft from parts (a) and (b)
(d) Answer: A union B = {4, 6, 8, 12, 16, 18, 20}
(e) B1 n((A union B)') = 13 cao, ft from part (d)
(e) Answer: n((A union B)') = 13

Question 3

M1 E = {2, 4, 6, 8, 10, 12} and M = {3, 6, 9, 12} both correctly identified (may be implicit)
A1 E intersection M = {6, 12} cao
Answer: E intersection M = {6, 12}

Question 4

(a) B1 Yes, because every element of B (4 and 8) is also an element of A
(a) Answer: Yes, B is a subset of A
(b) M1 attempts to list single-element subsets of B
(b) A1 {4} and {8} cao, both listed and no others
(b) Answer: {4} and {8}

Question 5

M1 at least 5 correct prime numbers less than 20 listed, with no non-prime numbers included
A1 P = {2, 3, 5, 7, 11, 13, 17, 19} cao, all 8 elements, none extra
Answer: P = {2, 3, 5, 7, 11, 13, 17, 19}

Question 6

B1 R intersection S = the empty set (or { }), cao
B1 correct reason, e.g. no number between 1 and 9 can be both even and odd, so R and S have no elements in common
Answer: R intersection S = the empty set

Question 7

B1 C cao
Answer: C

Question 8

(a) B1 A intersection B cao
(a) Answer: A intersection B
(b) B1 (A union B)' oe, e.g. A' intersection B'
(b) Answer: (A union B)'
(c) B1 A' intersection B cao
(c) Answer: A' intersection B

Question 9

M1 finds n(E intersection M) = 2, ft from Question 12
M1 uses n(E union M) = n(E) + n(M) - n(E intersection M), e.g. 6 + 4 - 2
A1 n(E union M) = 8 cao
Answer: n(E union M) = 8

Question 10

(a) M1 60 - 10, oe
(a) A1 n(F union N) = 50 cao
(a) Answer: n(F union N) = 50
(b) M1 uses n(F intersection N) = n(F) + n(N) - n(F union N), ft from part (a)
(b) A1 n(F intersection N) = 12 cao
(b) Answer: n(F intersection N) = 12
(c) B1 22 cao, ft from part (b): 34 - 12
(c) Answer: 22
(d) M1 finds Netball only = n(N) - n(F intersection N) = 28 - 12 = 16, ft, and forms 16/60
(d) A1 4/15 oe cao (simplest form)
(d) Answer: 4/15

Question 11

M1 quotes or uses n(A intersection B) = n(A) + n(B) - n(A union B)
M1 substitutes correctly, e.g. 22 + 19 - 35
A1 n(A intersection B) = 6 cao
Answer: n(A intersection B) = 6

Question 12

(a) M1 forms a correct equation by summing all eight regions to 100, e.g. 20 + 15 + 2x + x + 8 + 6 + 5 + 10 = 100
(a) M1 simplifies correctly, e.g. 64 + 3x = 100, leading to 3x = 36
(a) A1 x = 12 cao
(a) Answer: x = 12
(b) M1 identifies that n(S) = (S only) + (F and S only) + (G and S only) + (all three), ft their x
(b) M1 substitutes their x = 12 correctly, e.g. S only = 2(12) = 24
(b) A1 n(S) = 43 cao (24 + 8 + 6 + 5)
(b) Answer: n(S) = 43
(c) M1 identifies the correct four regions to add: all three, F and G only, F and S only, G and S only, ft their x
(c) A1 31 cao (5 + 12 + 8 + 6)
(c) Answer: 31
(d) M1 sums the three 'only' regions: 20 + 15 + their S only (24)
(d) A1 59/100 cao
(d) Answer: 59/100

Sets and Venn Notation - Worksheets, Questions and Revision

Related topics

1.1 Sets and Venn Notation

Question 1

Question 2

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Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

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Question 1

Question 2

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Question 11

Question 12