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Operations on Sets · Form 4

Operations on Sets: Practice Questions

Original SPM-style practice questions for Operations on Sets, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Operations on Sets, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

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

  1. A. {3, 5}
  2. B. {1, 2, 9, 11}
  3. C. {3, 5, 7}
  4. D. {1, 2, 3, 5, 7, 9, 11}

Question 2

Given P = {k, l, m} and Q = {l, m, n, o}, find P ∪ Q.

  1. A. {l, m}
  2. B. {k, l, m, n, o}
  3. C. {k, n, o}
  4. D. {k, l, m}

Question 3

The universal set is ξ = {x : 1 ≤ x ≤ 10, x is an integer}. If A = {2, 4, 6, 8, 10}, find A'.

  1. A. {2, 4, 6, 8, 10}
  2. B. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
  3. C. { }
  4. D. {1, 3, 5, 7, 9}

Question 4

Given n(A) = 18, n(B) = 12 and n(A ∩ B) = 5, find n(A ∪ B).

  1. A. 30
  2. B. 20
  3. C. 25
  4. D. 35

Question 5

In a Venn diagram, the shaded region contains all elements that are in set A but not in set B. Which notation represents the shaded region?

  1. A. A ∩ B'
  2. B. A' ∩ B
  3. C. A ∩ B
  4. D. (A ∪ B)'

Question 6

Given ξ = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find (A ∩ B)'.

  1. A. {3, 4}
  2. B. {1, 2, 5, 6, 7, 8}
  3. C. {7, 8}
  4. D. {1, 2, 5, 6}

Question 7

The shaded region of a Venn diagram represents (A ∪ B)'. By De Morgan's law, this is equal to which set?

  1. A. A' ∪ B'
  2. B. A ∩ B
  3. C. A' ∩ B'
  4. D. A ∪ B

Question 8

Given n(ξ) = 30 and n(A ∪ B) = 24, find n((A ∪ B)').

  1. A. 24
  2. B. 54
  3. C. 4
  4. D. 6

Question 9

If A ⊂ B, which of the following is equal to A ∩ B?

  1. A. A
  2. B. B
  3. C. ∅
  4. D. A ∪ B

Question 10

In a group of 50 students, 30 like tea and 25 like coffee, while 10 like both. How many students like neither drink?

  1. A. 45
  2. B. 5
  3. C. 10
  4. D. 15

Structured (Paper 2 style)

Question 1 (6 marks)

In a group of 60 students, 35 joined the Science Club (S), 28 joined the Mathematics Club (M) and 12 joined both clubs. (a) Find n(S ∪ M).

(b) Find the number of students who joined only one club. (c) Find the number of students who joined neither club.

  1. n(S ∪ M) = n(S) + n(M) − n(S ∩ M) = 35 + 28 − 12 = 51 students.
  2. Science only = 35 − 12 = 23; Mathematics only = 28 − 12 = 16.
  3. Students in only one club = 23 + 16 = 39 students.
  4. Students in neither club = n(ξ) − n(S ∪ M) = 60 − 51 = 9 students.

Question 2 (7 marks)

The universal set is ξ = {x : 1 ≤ x ≤ 15, x is an integer}. Set A is the set of multiples of 3 and set B is the set of factors of 12.

(a) List all elements of A and of B. (b) Find A ∩ B.

(c) Find A ∪ B. (d) Find (A ∪ B)'.

  1. Multiples of 3 up to 15: A = {3, 6, 9, 12, 15}.
  2. Factors of 12 within 1 to 15: B = {1, 2, 3, 4, 6, 12}.
  3. A ∩ B = common elements = {3, 6, 12}.
  4. A ∪ B = all elements listed once = {1, 2, 3, 4, 6, 9, 12, 15}.
  5. (A ∪ B)' = elements of ξ not in A ∪ B = {5, 7, 8, 10, 11, 13, 14}.

Question 3 (6 marks)

In a class of 40 students, 24 study Physics (P), 25 study Chemistry (C) and 6 study neither subject. Let x be the number who study both subjects.

(a) Form an equation in x and solve it to find n(P ∩ C). (b) Find the number who study Physics only.

(c) Find the number who study Chemistry only.

  1. Students studying at least one subject = 40 − 6 = 34, so n(P ∪ C) = 34.
  2. Using n(P ∪ C) = n(P) + n(C) − x: 34 = 24 + 25 − x.
  3. 34 = 49 − x, so x = 49 − 34 = 15; thus n(P ∩ C) = 15.
  4. Physics only = n(P) − x = 24 − 15 = 9.
  5. Chemistry only = n(C) − x = 25 − 15 = 10.

Question 4 (6 marks)

The universal set is ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 5}. (a) List A' and B'.

(b) Find (A ∩ B)'. (c) Find A' ∪ B' and hence verify that (A ∩ B)' = A' ∪ B'.

  1. A' = elements of ξ not in A = {1, 4, 6, 8, 9, 10}.
  2. B' = elements of ξ not in B = {6, 7, 8, 9, 10}.
  3. A ∩ B = {2, 3, 5}, so (A ∩ B)' = {1, 4, 6, 7, 8, 9, 10}.
  4. A' ∪ B' = {1, 4, 6, 8, 9, 10} ∪ {6, 7, 8, 9, 10} = {1, 4, 6, 7, 8, 9, 10}.
  5. Both results equal {1, 4, 6, 7, 8, 9, 10}, verifying De Morgan's law (A ∩ B)' = A' ∪ B'.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What's the difference between the Paper 1 and Paper 2 questions in this sets practice set?

Paper 1 questions are multiple-choice, checking quick recognition of union, intersection, and complement notation. Paper 2 questions ask you to draw and shade Venn diagrams, list set elements, and explain your reasoning in full steps, practising the working that Paper 2 marks reward.

Why attempt each set question myself before checking the worked solution?

Shading a Venn diagram correctly takes practice reading which regions a question actually describes. Attempting it yourself first shows whether you can translate set notation into a diagram under exam time pressure, rather than just recognising a correct diagram once it's drawn for you.

What common mistakes appear in operations on sets questions?

Students often shade the wrong region for a complement, confuse ∩ with ∪, or forget elements shared between two sets when listing them once. Working through this practice set helps you catch these slips before they cost marks in the real exam.

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