Sets: subsets, set-builder notation and Venn problems

Sets can be described by a rule, can sit inside each other, and can be counted with a Venn diagram even when a number is unknown.

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Subsets and the empty set

A ⊂ B means every member of A is also in B.
{2, 4} ⊂ {1, 2, 3, 4}
∅ is the empty set, with no members. A ∩ B = ∅ means A and B have nothing in common.
n(A) is the number of members of A.

Set-builder notation

{x : x is an integer, −2 ≤ x < 3} means all integers x from −2 up to, but not including, 3.
So it is {−2, −1, 0, 1, 2}.
{x : x > 5} is every number greater than 5.

Venn diagrams with an unknown

n(ξ) = 40, n(A) = 22, n(B) = 15 and 8 are in neither. Let n(A ∩ B) = x.
A only = 22 − x, B only = 15 − x.
(22 − x) + x + (15 − x) + 8 = 40
45 − x = 40, so x = 5.
Worked example

In a class of 40, 22 study French, 15 study Spanish and 8 study neither. How many study both?

  1. Let x study both. French only = 22 − x, Spanish only = 15 − x.
  2. (22 − x) + x + (15 − x) + 8 = 40
  3. 45 − x = 40, so x = 5

Answer: 5 students

Key idea

A ⊂ B: everything in A is in B. ∅ is the empty set. Set-builder notation gives the rule. With an unknown overlap, add up all the regions and set the total equal to n(ξ).

Key words
Venn diagram
Overlapping circles inside a rectangle that show sets and how they overlap.
Integer
A whole number: positive, negative or zero.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. B = {1, 3, 5, 7, 9}. Which set is a subset of B?

    1. {1, 2}
    2. {3, 9}
    3. {5, 7, 11}
    4. {0, 1}
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    Answer: {3, 9}

    Every member of {3, 9} is in B.

  2. n(A) = 40, n(B) = 16 and n(A ∩ B) = 9. Find n(A ∪ B).

    Show the answer

    Answer: 47

    n(A ∪ B) = 40 + 16 − 9 = 47. The overlap was counted twice.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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