Mathematics questions, chapter 4

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Operations on Sets

44 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Do not use a calculator. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer.

Question 1

Which symbol represents the intersection of sets?

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∩ is the intersection (read "and"). ∪ is the union (read "or"), ′ is the complement and ⊂ means "is a subset of".

For the teacher: Remembering, low difficulty, standard 4.1.1

Textbook p. 96

Question 2

What does the set P ∪ Q contain?

Only the elements that are in both P and Q
Only the elements that are in P but not in Q
All the elements that are in P or in Q or in both
All the elements of ξ that are not in P or Q

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The union contains every element of P, every element of Q, and the common elements once. The first option describes P ∩ Q and the last describes (P ∪ Q)′.

For the teacher: Understanding, low difficulty, standard 4.2.1

Textbook p. 106

Question 3

A = {1, 3, 5, 7, 9, 11} and B = {2, 3, 5, 7, 11}. Find A ∩ B.

{3, 5, 7, 11}
{1, 2, 3, 5, 7, 9, 11}
{1, 9}
{2}

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The elements in both sets are 3, 5, 7 and 11. The second option is A ∪ B. {1, 9} is the part of A that is not in B, and {2} is the part of B that is not in A.

For the teacher: Applying, low difficulty, standard 4.1.1

Textbook p. 98

Question 4

P = {a, b, c, d} and Q = {c, d, e}. Find n(P ∪ Q).

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P ∪ Q = {a, b, c, d, e}, so n(P ∪ Q) = 5. Check: 4 + 3 − 2 = 5. 7 counts c and d twice, and 2 is n(P ∩ Q).

For the teacher: Applying, low difficulty, standard 4.2.1

Textbook p. 108

Question 5

ξ = {1, 2, 3, …, 10}, A = {2, 4, 6, 8, 10} and B = {4, 8}. List the elements of (A ∩ B)′.

{4, 8}
{1, 3, 5, 7, 9}
{1, 2, 3, 5, 6, 7, 9, 10}
{2, 6, 10}

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A ∩ B = {4, 8}, so (A ∩ B)′ is everything else in ξ: {1, 2, 3, 5, 6, 7, 9, 10}. {4, 8} is A ∩ B itself, {1, 3, 5, 7, 9} is A′ and {2, 6, 10} is in A but not B.

For the teacher: Applying, medium difficulty, standard 4.1.2

Textbook p. 100

Question 6

ξ = {1, 2, 3, …, 10}, P = {1, 2, 3, 4} and Q = {3, 4, 5, 6}. Find n((P ∪ Q)′).

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P ∪ Q = {1, 2, 3, 4, 5, 6}, so (P ∪ Q)′ = {7, 8, 9, 10} and n = 4. 6 is n(P ∪ Q) and 2 is n(P ∩ Q).

For the teacher: Applying, medium difficulty, standard 4.2.2

Textbook p. 110

Question 7

If P ∩ Q = φ, which statement about sets P and Q is true?

P = Q
P ⊂ Q
P ∪ Q = φ
P and Q have no element in common

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P ∩ Q = φ means the intersection is empty, so no element is in both sets. P and Q may still have elements of their own, so P ∪ Q need not be empty.

For the teacher: Understanding, medium difficulty, standard 4.1.1

Textbook p. 99

Question 8

ξ = {1, 2, 3, …, 12}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and C = {4, 6, 8, 10}. List the elements of (A ∪ B) ∩ C.

{4}
{4, 6}
{1, 2, 3, 4, 5, 6, 8, 10}
{3, 4, 6}

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A ∪ B = {1, 2, 3, 4, 5, 6}. Of these, only 4 and 6 are in C. {4} is A ∩ B ∩ C, and the third option is A ∪ B ∪ C.

For the teacher: Applying, medium difficulty, standard 4.3.1

Textbook p. 116

Question 9

Diagram 1 is a Venn diagram showing sets P, Q and R. Which set is represented by the shaded region?

Venn diagram of P, Q and R; the part of R that lies inside P or inside Q is shaded — Diagram 1

P ∩ Q ∩ R
P ∪ (Q ∩ R)
(P ∪ Q) ∩ R
(P ∩ Q) ∪ R

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Only the part of R that is also in P or in Q is shaded, which is (P ∪ Q) ∩ R. P ∩ Q ∩ R would be just the centre. P ∪ (Q ∩ R) and (P ∩ Q) ∪ R would shade the whole of P or the whole of R.

For the teacher: Analysing, medium difficulty, standard 4.3.1

Textbook p. 117

Question 10

There are 50 pupils in a class. 24 pupils join the Chess Club and 20 pupils join the Drama Club. 8 pupils join both clubs. How many pupils join neither club?

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n(Chess ∪ Drama) = 24 + 20 − 8 = 36, so neither = 50 − 36 = 14. 6 forgets the overlap (50 − 44). 36 is the number who join at least one club.

For the teacher: Applying, medium difficulty, standard 4.2.3

Textbook p. 113

Question 11

Diagram 2 shows the number of elements in sets A and B. n(ξ) = 32. Find the value of x.

Venn diagram: A only 2x, A and B x, B only x + 3, outside both sets 5 — Diagram 2

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2x + x + (x + 3) + 5 = 32, so 4x + 8 = 32 and 4x = 24, giving x = 6. Check: 12 + 6 + 9 + 5 = 32.

For the teacher: Analysing, high difficulty, standard 4.2.3

Textbook p. 115

Question 12

Diagram 3 shows the number of elements in each region of sets A, B and C. Find n(A′ ∩ (B ∪ C)).

Venn diagram: A only 5, B only 7, C only 6, A and B only 3, B and C only 2, A and C only 4, all three 1, outside all 8 — Diagram 3

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A′ ∩ (B ∪ C) is the part of B or C that is outside A: B only 7 + C only 6 + (B and C only) 2 = 15. 23 is n(B ∪ C), which includes the parts inside A. 16 wrongly adds the centre (1) as well.

For the teacher: Analysing, high difficulty, standard 4.3.3

Textbook p. 121

Section B: structured questions

32 marks

Answer all questions. Show your working.

Question 13

ξ = {x : x is an integer, 1 ≤ x ≤ 12}, P = {x : x is an even number} and Q = {x : x is a factor of 12}.

[2 marks] (a)List the elements of P and of Q.

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P = {2, 4, 6, 8, 10, 12} [1]1
Q = {1, 2, 3, 4, 6, 12} [1]1

Tick each point you got.

[1 mark] (b)List the elements of P ∩ Q.

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P ∩ Q = {2, 4, 6, 12} [1]1

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[1 mark] (c)Find n(P ∪ Q).

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P ∪ Q = {1, 2, 3, 4, 6, 8, 10, 12}, n(P ∪ Q) = 8 [1]1

Check: 6 + 6 − 4 = 8.

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 4.1.1

Textbook p. 98

Question 14

ξ = {x : x is an integer, 20 ≤ x ≤ 30}, G = {x : x is a prime number}, H = {x : x is a multiple of 4} and I = {x : x is a multiple of 5}.

[2 marks] (a)List the elements of (G ∪ H)′.

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G = {23, 29}, H = {20, 24, 28}, so G ∪ H = {20, 23, 24, 28, 29} [1]1
(G ∪ H)′ = {21, 22, 25, 26, 27, 30} [1]1

Tick each point you got.

[1 mark] (b)Find n((H ∪ I)′).

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H ∪ I = {20, 24, 25, 28, 30}, so n((H ∪ I)′) = 11 − 5 = 6 [1]1

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[1 mark] (c)(G ∩ H)′ = ξ

ATrueBFalse

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True

G and H have no element in common, so G ∩ H = φ and its complement is the whole universal set.

For the teacher: Applying, medium difficulty, standard 4.2.2

Textbook p. 110

Question 15

Diagram 4 is a Venn diagram showing ξ = {1, 2, 3, …, 12}, A = {1, 2, 3, 4, 6, 12}, B = {2, 4, 6, 8, 10, 12} and C = {3, 6, 9, 12}.

Venn diagram of xi = 1 to 12 with A, B and C; 6 and 12 are in all three sets; 2 and 4 are in A and B only; 3 is in A and C only; 1 in A only; 8 and 10 in B only; 9 in C only; 5, 7, 11 outside all sets — Diagram 4

[1 mark] (a)List the elements of (A ∪ B) ∩ C.

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(A ∪ B) ∩ C = {3, 6, 12} [1]1

Tick each point you got.

[2 marks] (b)List the elements of A′ ∩ (B ∪ C).

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A′ = {5, 7, 8, 9, 10, 11} and B ∪ C = {2, 3, 4, 6, 8, 9, 10, 12} [1]1
A′ ∩ (B ∪ C) = {8, 9, 10} [1]1

Tick each point you got.

[1 mark] (c)On a copy of Diagram 4, shade the region that represents (A ∪ B) ∩ C.

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Venn diagram: the regions containing 3, 6 and 12, inside C and also inside A or B, are shaded

The part of C that lies inside A or inside B is shaded (the regions with 3, 6 and 12) and 9 is left unshaded [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 4.3.2

Textbook p. 118

Question 16

A total of 60 pupils are asked about their sport. 38 pupils swim (S), 29 pupils cycle (C) and 12 pupils do both.

[1 mark] (a)How many pupils swim only?

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38 − 12 = 26 [1]1

Tick each point you got.

[1 mark] (b)How many pupils cycle only?

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29 − 12 = 17 [1]1

Tick each point you got.

[2 marks] (c)How many pupils do neither sport? Draw a Venn diagram to help you.

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Venn diagram: swimming only 26, both 12, cycling only 17, neither 5

n(S ∪ C) = 26 + 12 + 17 = 55 [1]1
60 − 55 = 5 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 4.2.3

Textbook p. 113

Question 17

A total of 80 pupils choose clubs. 30 pupils join Chess (H), 25 pupils join Drama (D) and 28 pupils join Robotics (R). 10 pupils join Chess and Drama, 8 pupils join Drama and Robotics and 9 pupils join Chess and Robotics (each of these three numbers includes the pupils in all three clubs). 4 pupils join all three clubs.

[2 marks] (a)How many pupils join exactly two clubs?

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Chess and Drama only = 10 − 4 = 6; Drama and Robotics only = 8 − 4 = 4; Chess and Robotics only = 9 − 4 = 5 [1]1
6 + 4 + 5 = 15 [1]1

Tick each point you got.

[2 marks] (b)How many pupils join at least one club?

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Venn diagram: Chess only 15, Drama only 11, Robotics only 15, Chess and Drama only 6, Drama and Robotics only 4, Chess and Robotics only 5, all three 4, none 20

Chess only = 30 − 6 − 5 − 4 = 15, Drama only = 25 − 6 − 4 − 4 = 11, Robotics only = 28 − 4 − 5 − 4 = 15 [1]1
15 + 11 + 15 + 15 + 4 = 60 [1]1

Tick each point you got.

[1 mark] (c)How many pupils join no club?

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80 − 60 = 20 [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, standard 4.3.3

Textbook p. 121

Question 18

Diagram 5 shows sets P and Q in a universal set ξ with n(ξ) = 40. It is given that n(P ∩ Q) = n((P ∪ Q)′) = k.

Venn diagram: P only 10, P and Q k, Q only 6, outside both sets k — Diagram 5

[2 marks] (a)Form an equation in k and find its value.

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10 + k + 6 + k = 40 [1]1
2k = 24, so k = 12 [1]1

Tick each point you got.

[1 mark] (b)Find n(P).

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n(P) = 10 + 12 = 22 [1]1

Tick each point you got.

[1 mark] (c)Find n(Q′).

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n(Q) = 6 + 12 = 18, so n(Q′) = 40 − 18 = 22 [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 4.2.3

Textbook p. 115

Question 19

ξ = {1, 2, 3, …, 10}, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8, 10}. Diagram 6 shows Hafiz's working to find n(A′ ∩ B).

Hafiz's working. Line 1: A' = {7, 8, 9, 10}. Line 2: A' ∩ B = {2, 4, 6, 7, 8, 9, 10}. Line 3: n(A' ∩ B) = 7 — Diagram 6

[2 marks] (a)In which line did Hafiz make his first mistake? Explain the mistake.

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Line 2 [1]1
He listed the elements that are in A′ or in B (the union), but ∩ needs the elements in both A′ and B [1]1

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[1 mark] (b)Write the correct A′ ∩ B and find n(A′ ∩ B).

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A′ ∩ B = {8, 10}, n(A′ ∩ B) = 2 [1]1

Tick each point you got.

For the teacher: Evaluating, medium difficulty, standard 4.3.2

Textbook p. 118

Question 20

ξ = {1, 2, 3, …, 12}. Design two sets A and B such that n(A) = 5, n(B) = 5, n(A ∩ B) = 2 and n((A ∪ B)′) = 4.

[2 marks] (a)List the elements of your sets A and B. Show that n((A ∪ B)′) = 4.

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A and B each with 5 elements of ξ sharing exactly 2, e.g. A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8} [1]1
A ∪ B has 5 + 5 − 2 = 8 elements, so n((A ∪ B)′) = 12 − 8 = 4 [1]1

Tick each point you got.

[2 marks] (b)Draw a Venn diagram to represent ξ, A and B.

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Sample answer (other correct sets are accepted):
Venn diagram: A only 1, 2, 3; overlap 4, 5; B only 6, 7, 8; outside both sets 9, 10, 11, 12

A and B overlapping inside the rectangle ξ, with the 2 common elements in the overlap [1]1
All 12 elements in the correct regions, with 4 outside both sets [1]1

Tick each point you got.

For the teacher: Creating, high difficulty, standard 4.3.3

Textbook p. 121

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