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Grade 5 · Foundation and Higher · free GCSE maths worksheet

Set notation: subsets and n(A)

Edexcel IGCSE

Subsets, n(A), the empty and universal sets, complements and sets described by a rule.

10 original questions, 34 marks. Type your answers below for an instant mark and a score, or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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How to do it

  1. Write every set out in full first: turn the rule into a list of members
  2. ∩ means in both, ∪ means in either, A' means in E but not in A
  3. n(A) is just a count: how many members A has
  4. A ⊂ B is true only if every member of A is also in B
  5. n(A ∪ B) = n(A) + n(B) - n(A ∩ B): take off the overlap you counted twice

Watch out: A' is everything in the universal set E that is not in A, so check what E is before you write a complement.

Worked example

ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {x : x is an integer and 3 ≤ x < 9}

B = {even numbers}

  1. List the members of A ∩ B′.
  2. Find n((A ∪ B)′).

Write the sets out in full first

A = {3, 4, 5, 6, 7, 8}

B = {2, 4, 6, 8, 10}

B' = {1, 3, 5, 7, 9}

a) in A and in B'

A ∩ B' = {3, 5, 7}

b) in A or in B, then everything else

A ∪ B = {2, 3, 4, 5, 6, 7, 8, 10}

(A ∪ B)' = {1, 9}

n((A ∪ B)') = 2

The questions

1

ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {multiples of 3}

  1. List the members of A.
  2. Is 8 ∈ A? Write yes or no.
  3. Write down n(A).
3 marks
Show answer
a) {3, 6, 9, 12} b) no c) 4
2

ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

S = {square numbers}

  1. List the members of S.
  2. List the members of S′.
  3. Write down n(S′).
3 marks
Show answer
a) {1, 4, 9} b) {2, 3, 5, 6, 7, 8, 10} c) 7
3

A = {x : x is an integer and 2 < x ≤ 7}

B = {x : x2 < 20, x is an integer}

  1. List the members of A.
  2. List the members of B.
  3. Find n(A ∩ B).
3 marks
Show answer
a) {3, 4, 5, 6, 7} b) {-4, -3, -2, -1, 0, 1, 2, 3, 4} c) 2
4

ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

P = {prime numbers}

V = {even numbers}

  1. List the members of P ∩ V.
  2. List the members of P ∩ V′.
  3. List the members of (P ∪ V)′.
3 marks
Show answer
a) {2} b) {3, 5, 7, 11} c) {1, 9}
5

S = {3, 7, 9}

List all the subsets of S. Remember that ∅ and S itself are both subsets of S.

3 marks
Show answer
∅, {3}, {7}, {9}, {3, 7}, {3, 9}, {7, 9}, {3, 7, 9} (8 subsets)
6

ℰ = {integers from 1 to 20}

A = {multiples of 4}

B = {even numbers}

C = {prime numbers}

For each statement, write yes if it is true or no if it is false.

  1. A ⊂ B
  2. A ∩ C = ∅
  3. C ⊂ B′
3 marks
Show answer
a) yes b) yes c) no
7

ℰ = {x : x is an integer and 1 ≤ x ≤ 15}

A = {x : x is a multiple of 3}

B = {x : x > 9}

  1. Find n(A).
  2. List the members of A ∩ B.
  3. Find n(A′).
  4. Find n((A ∪ B)′).
4 marks
Show answer
a) 5 b) {12, 15} c) 10 d) 6
8

There are 32 students in a Year 10 class.

F is the set of students who study French and S is the set of students who study Spanish.

n(F) = 19, n(S) = 14 and n(F ∩ S) = 8.

  1. Work out n(F ∪ S).
  2. Work out how many students study neither French nor Spanish.
3 marks
Show answer
a) 25 b) 7
9

60 people were asked which parts of a leisure centre they used last week.

P is the set of people who used the pool and C is the set of people who used the climbing wall.

38 people used the pool, 29 people used the climbing wall and 9 people used neither.

  1. Find n(P ∩ C).
  2. Find n(P ∩ C′).
4 marks
Show answer
a) 16 b) 22
10

ℰ = {x : x is an integer and −5 ≤ x ≤ 5}

A = {x : x2 ≥ 10}

B = {x : 2x + 1 > 0}

C = {x : x2 = 16}

  1. List the members of A.
  2. Find n(B).
  3. List the members of A ∩ B′.
  4. Is C ⊂ A? Write yes or no.
5 marks
Show answer
a) {-5, -4, 4, 5} b) 6 c) {-5, -4} d) yes

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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