Venn diagram rules

The main Venn diagram rules are: the rectangle is the universal set, ξ, containing everything being considered; each circle is a set; the overlap of two circles is the intersection, A ∩ B, things in both; everything inside either circle is the union, A ∪ B; and everything outside a circle is its complement, A′. Every item goes in exactly one region, so the numbers in all the regions add up to the total. When filling one in, always start with the overlap in the middle.

Venn diagram notation

A memory aid: ∩ looks like a lowercase n, for "and"; ∪ looks like a u, for "union", which means "or". AQA's GCSE Maths specification asks students to enumerate sets and combinations of sets systematically using Venn diagrams, among other methods.

How to fill in a Venn diagram

1. Draw the rectangle and the circles, and label them.
2. Put the number in both sets in the overlap first.
3. For each circle, subtract the overlap from that set's total to get the "only" region.
4. Subtract everything inside the circles from the overall total to get the number outside both.
5. Check that all the regions add up to the total.

Example 1: filling in from totals

In a class of 30 students, 18 play football (F), 12 play tennis (T), and 5 play both. Draw a Venn diagram.

Overlap: 5. Football only: 18 − 5 = 13. Tennis only: 12 − 5 = 7. Neither: 30 − (13 + 5 + 7) = 5.

A common mistake is to write 18 in the football-only region. The 18 who play football include the 5 who also play tennis, so only 13 go in the part of F outside the overlap.

Probability from a Venn diagram

Divide the number in the region you want by the total:

• P(F) = 18 ÷ 30
• P(F ∩ T), playing both = 5 ÷ 30
• P(F ∪ T), playing at least one = (13 + 5 + 7) ÷ 30 = 25 ÷ 30
• P(neither) = 5 ÷ 30

The addition rule

Adding P(F) and P(T) counts the overlap twice, so subtract it once:

Check with Example 1: 18/30 + 12/30 − 5/30 = 25/30, matching the count. If the circles do not overlap, the events are mutually exclusive, P(A ∩ B) = 0, and the rule becomes P(A ∪ B) = P(A) + P(B).

Conditional probability from a Venn diagram

If you are told something is already in one set, the total shrinks to that set. Given that a student plays tennis, the probability that they also play football is:

AQA's specification includes calculating and interpreting conditional probabilities using Venn diagrams, in its Higher content.

Now try some OR and AND problems in the course, then press Mark to check them there.

Venn diagrams for HCF and LCM

Venn diagrams also find the highest common factor (HCF) and lowest common multiple (LCM) of two numbers. Write each number as a product of primes, put the shared primes in the overlap and the rest in their own circles.

Example 2: HCF and LCM of 60 and 84

60 = 2 × 2 × 3 × 5 and 84 = 2 × 2 × 3 × 7. They share 2, 2 and 3.

• HCF = product of the primes in the overlap: 2 × 2 × 3 = 12.
• LCM = product of all the primes in the diagram: 5 × 2 × 2 × 3 × 7 = 420.

The LCM Calculator and GCF Calculator show the prime factor working, and our guide to prime factor decomposition covers splitting numbers into primes.

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LCM Calculator

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Three-circle Venn diagrams

The same rules work with three sets. Start with the centre, where all three circles overlap, then fill the regions where exactly two overlap (subtracting the centre each time), then the "only" regions, and finally the outside. There are eight regions in all, including the one outside every circle.

Common mistakes

• Not subtracting the overlap. The "only" region is the set's total minus the overlap.
• Forgetting the outside. Items in neither set still go in the rectangle.
• Mixing up ∪ and ∩. ∪ is "or" (either circle); ∩ is "and" (the overlap).
• Using the whole total for conditional probability. "Given that" means divide by the size of that set only.
• Adding P(A) and P(B) when the sets overlap. Subtract P(A ∩ B) once.

Practise probability in a short timed paper

GCSE Maths Higher and GCSE Maths Foundation each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so Venn diagrams and probability turn up regularly but not in every paper. The Probability course has an interactive Venn diagram builder.

Read next

Maths · ProbabilityConditional probability questions with worked answersConditional probability questions from two-way tables, Venn diagrams and tree diagrams, with the formula, full worked answers and practice questions.