GCSE Set Notation & Venn Diagrams | The Maths and Science Tutor

Set notation made its way fully onto the GCSE syllabus in 2017 and has been a regular feature ever since — usually six to eight marks across two parts on a Higher paper. The symbols look unfamiliar at first but encode three simple ideas. Once you've drilled the notation and the shading, the probability questions that follow are mechanical.

The symbols you need

ξ — the universal set (everything we're considering)
A ∪ B — the union of A and B (everything in A, or B, or both)
A ∩ B — the intersection of A and B (everything in both)
A' — the complement of A (everything NOT in A)
n(A) — the number of elements in A
x ∈ A — x is an element of A
∅ — the empty set

Shading regions on a Venn diagram

The exam question often shows a blank Venn diagram with two or three circles and asks you to shade a particular region.

A ∪ B — shade everything inside either circle
A ∩ B — shade only the overlap
A' — shade everything outside A (including the rectangle around the circles)
(A ∪ B)' — shade everything outside both circles
A ∩ B' — shade A but not the part that overlaps with B

Examiner's note: Read combined symbols left to right but apply complement (the dash) last to the whole expression in brackets. (A ∪ B)' means "first work out A ∪ B, then take the complement of that".

Filling in numbers — the standard exam technique

Worked example 1

50 students. 30 study French. 25 study German. 18 study both. Draw a Venn diagram and find n(F ∪ G).

Start with the intersection: 18 students study both. Put 18 in the overlap.

French only: 30 − 18 = 12. German only: 25 − 18 = 7.

Outside both: 50 − (12 + 18 + 7) = 13.

n(F ∪ G) = 12 + 18 + 7 = 37

Examiner's note: Always start with the overlap and work outwards. Trying to fill in "French" first (and then forgetting that it includes the overlap students) is the single most common mistake on this topic.

Probability from a Venn diagram

Once the diagram is filled in, probabilities are just ratios.

Worked example 2 — using the data above

P(student studies French) = 30 / 50 = 3/5

P(student studies both) = 18 / 50 = 9/25

P(student studies neither) = 13 / 50

P(student studies French given they study German) = 18 / 25 (the conditional)

The mistakes that cost the most marks

Mistake 1 — Filling in "French = 30" inside the French-only part. 30 includes the students who also do German. Always start with the overlap and subtract outwards.

Mistake 2 — Forgetting the universal set. Some students appear in neither circle. The numbers in all regions must add up to the total.

Mistake 3 — Confusing union and intersection symbols. ∪ looks like a "u" and is "or" (union). ∩ is the upside-down ∪ and is "and" (intersection). Get the symbols mixed up and the whole answer collapses.

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Set Notation & Venn Diagrams