Read and write set notation, list the members of sets, use n(A), and describe sets with symbols. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Objectives
Read the symbols ∈, ∪, ∩, A' and n(A).
List the members of sets such as multiples, factors, primes and squares.
Find a union, an intersection and a complement.
Key points
Write a set in curly brackets, such as {2, 4, 6, 8}, and say whether a number belongs to it using ∈ and ∈/.
Use ∪ for union (in either), ∩ for intersection (in both), A′ for not in A, and ξ for the universal set.
Use n(A) for the number of members of a set, and ∅ for the empty set.
Use A⊂B when every member of A is also in B.
Worked examples
Example 1
ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}. Write down n(A').
2 marks
A' is everything in ξ that is not in A
A' = {1, 3, 5, 7, 9}
Count the members: n(A') = 5
Answer: 5
Common slips
Slip: A' is the same as A The complement is the members left out of A.
Slip: A' = {2, 4, 6, 8, 10} Cross out the members of A.
Slip: n(A') = 10 Count the members of A', not of ξ.
Example 2
ξ = {1, 2, ..., 12}. A is the set of factors of 12 and B is the set of even numbers. Work out n(A ∪ B).
2 marks
A = {1, 2, 3, 4, 6, 12} and B = {2, 4, 6, 8, 10, 12}
A ∩ B = {2, 4, 6, 12}, so n(A ∩ B) = 4
n(A ∪ B) = 6 + 6 - 4 = 8
Answer: 8
Common slips
Slip: A = {2, 3, 4, 6, 12} and B = {2, 4, 6, 8, 10} A factor list includes 1 and 12 itself, and the even numbers include 12.
Slip: A ∩ B = {1, 2, 3, 4, 6, 8, 10, 12} The intersection is the members in both lists.
Slip: n(A ∪ B) = 6 + 6 = 12 The members in both sets are counted twice.
Example 3
ξ is the whole numbers from 1 to 20. P is the set of primes and S is the set of squares. One number is chosen at random. Find the probability that it is in neither P nor S.
3 marks
P = {2, 3, 5, 7, 11, 13, 17, 19}, so n(P) = 8
S = {1, 4, 9, 16}, so n(S) = 4
No number is in both lists, so n(P ∪ S) = 8 + 4 = 12
The numbers in neither set are 20 - 12 = 8
The probability is 8/20 = 2/5
Answer: 2/5
Common slips
Slip: P = {1, 2, 3, 5, 7, 11, 13, 17, 19} 1 is not a prime number.
Slip: S = {4, 9, 16} 1 is a square number.
Slip: n(P ∪ S) = 8 Add the squares as well.
Slip: The numbers in neither set are 12 12 is the number in a set, not outside.
Slip: The probability is 8/12 The total for the probability is all 20 numbers.
Questions and answers
GCSE Maths sets and set notation questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
List the members of the set of multiples of 4 that are less than 20.
Show answer
Answer: {4, 8, 12, 16}
4, 8, 12, 16 are multiples of 4.
20 is not less than 20.
M1 — 4, 8, 12, 16 are multiples of 4.
A1 — 20 is not less than 20.
Question 2 Easy2 marks
ξ = {1, 2, 3, 4, 5, 6}. A = {1, 3, 5}. List A'.
Show answer
Answer: {2, 4, 6}
A' is in ξ but not in A.
The members left are 2, 4 and 6.
M1 — A' is in ξ but not in A.
A1 — The members left are 2, 4 and 6.
Question 3 Medium2 marks
A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}. List A ∩ B.
Show answer
Answer: {4, 5}
The intersection keeps members in both lists.
4 and 5 are in both.
M1 — The intersection keeps members in both lists.
A1 — 4 and 5 are in both.
Question 4 Medium2 marks
A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}. Work out n(A ∪ B).
Show answer
Answer: 7
The union is {1, 2, 3, 4, 5, 6, 7}.
There are 7 members.
M1 — The union is {1, 2, 3, 4, 5, 6, 7}.
A1 — There are 7 members.
Question 5 Hard2 marks
n(A) = 9, n(B) = 7 and n(A ∩ B) = 3. Work out n(A ∪ B).
Show answer
Answer: 13
Add the counts and take away the overlap.
9 + 7 - 3 = 13.
M1 — Add the counts and take away the overlap.
A1 — 9 + 7 - 3 = 13.
Question 6 Hard2 marks
List the members of the set {x : x is an integer, 1 < x ≤ 5}.
Show answer
Answer: {2, 3, 4, 5}
1 < x means 1 is left out.
x ≤ 5 means 5 is included.
M1 — 1 < x means 1 is left out.
A1 — x ≤ 5 means 5 is included.
Question 7 Exam-style4 marks
ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. A = {x : x is a multiple of 3} and B = {x : x is a factor of 12}. List the members of A ∩ B. Work out n(B').
Show answer
Answer: {3, 6, 12} and 6
A = {3, 6, 9, 12} and B = {1, 2, 3, 4, 6, 12}.
A ∩ B = {3, 6, 12}.
n(B') = 12 - 6 = 6.
M1 — Lists A and B correctly
A1 — {3, 6, 12}
M1 — Uses 12 - n(B) or lists B'
A1 — 6
Common mistakes
Most lost marks on this topic are one slip.
Mixing up ∩ and ∪
The cap ∩ is in both, the cup ∪ is in either
Writing n(A ∪ B) as n(A) + n(B)
Take away the members counted twice, n(A ∩ B)
Finding A' from another set instead of ξ
List ξ, then take away the members of A
Listing 1 as a prime number
A prime has exactly two factors, so the primes start at 2
Spot the slip
A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Ben says A ∩ B = {1, 2, 3, 4, 5, 6}. Which set did he write down?
The union A ∪ B
The intersection A ∩ B
The complement A'
Show the slip
The union A ∪ B He wrote the union, everything in either set. The intersection is the members in both, which is {3, 4}.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.