Skip to content
VirtusAcademy

Venn Diagrams

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on Venn diagrams helps you revise using Venn diagrams and set notation. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Fill in the overlap first, then work outwards.

Free downloads

These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A Venn diagram shows how sets overlap, using circles inside a rectangle that represents everything under consideration.

The overlap contains items belonging to both sets, and the region outside the circles contains items in neither. Filling the diagram always starts from the intersection and works outwards, because items in the overlap would otherwise be counted twice.

Probabilities are read by dividing the relevant region's count by the total. The notation matters: \(A \cap B\) means both, \(A \cup B\) means either or both, and \(A'\) means not \(A\).

Revision notes

Filling in the diagram

Start with the intersection, then subtract it from each set total to find the parts that belong to only one set.

If \(20\) like maths, \(15\) like science and \(8\) like both, then \(12\) like only maths and \(7\) like only science.

The notation

\(A \cap B\) is the intersection, meaning both. \(A \cup B\) is the union, meaning either or both.

\(A'\) is the complement, meaning everything not in \(A\). The rectangle holds the universal set, everything under consideration.

Reading probabilities

Count the items in the region you want and divide by the overall total.

For \(P(A \cap B)\), use only the overlap. For \(P(A \cup B)\), use everything inside either circle.

Key points

  • A Venn diagram shows overlapping sets.
  • The overlap contains items in both sets.
  • Start filling from the intersection.
  • \(A \cap B\) means both.
  • \(A \cup B\) means either or both.
  • \(A'\) means not in \(A\).

Worked examples

Example 1

\(20\) like tea, \(14\) like coffee, \(6\) like both. How many like only tea?

Working

\[20 - 6\]subtract the overlap from the tea total
\[= 14\]state the number who like only tea

Example 2

Using those figures with \(30\) people surveyed, find \(P(\text{both})\).

Working

\[\frac{6}{30}\]the overlap over the total
\[= \frac{1}{5}\]simplify the fraction

Example 3

With \(20\) tea, \(14\) coffee and \(6\) both, how many like at least one?

Working

\[20 + 14 - 6\]add the sets and subtract the double-counted overlap
\[= 28\]state the union

Common mistakes

  • Not subtracting the overlap.

    Items in both sets would otherwise be counted twice.

  • Filling the outer regions first.

    Start from the intersection and work outwards.

  • Confusing union with intersection.

    Union means either or both; intersection means both.

  • Forgetting the region outside the circles.

    Items in neither set still count towards the total.

Exam tips

  • Always start filling from the intersection.
  • Check all the regions add to the overall total.
  • Learn the union, intersection and complement notation.
  • Divide by the overall total unless the question restricts it.

Key terms

Intersection
The region belonging to both sets.
Union
The region belonging to either set or both.
Complement
Everything not in a given set.
Universal set
Everything under consideration, shown by the rectangle.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.