Venn diagrams, explained with examples
A Venn diagram uses overlapping circles to sort items into groups. In GCSE Maths, Venn diagrams organise information, use set notation and make probability questions much easier.
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The parts of a Venn diagram
The rectangle represents the universal set, written ξ (the Greek letter xi), which contains everything being considered. Each circle is a set. Where circles overlap, items belong to both sets. Items outside every circle, but inside the rectangle, belong to none of them. A number in a region usually means how many items are in that region; sometimes the actual items are listed instead.
How to fill in a Venn diagram
The golden rule is to start in the middle. Fill the overlap first, then work outwards.
- Put the number in both sets into the overlap.
- Subtract it from each set's total to get the “only” regions.
- Subtract everything inside the circles from the total to get the number outside.
- Check that all the regions add up to the total. If they do not, recheck each subtraction before going on.
Example 1: filling a diagram
In a class of 30, 14 play football, 11 play tennis and 4 play both. How many play neither?
- Overlap: 4. Football only: 14 − 4 = 10. Tennis only: 11 − 4 = 7.
- Inside the circles: 10 + 4 + 7 = 21. Neither: 30 − 21 = 9. This is the diagram above.
- A common mistake is to add 14 + 11 = 25 and get 5 for neither, which counts the 4 students who play both twice.
Set notation
- A ∩ B (A intersection B): in A and B, the overlap.
- A ∪ B (A union B): in A or B or both, everything inside either circle.
- A′ (A complement): not in A, everything outside circle A.
- n(A): the number of items in A.
- x ∈ A: x is a member of A.
Memory aids: ∩ looks like an n for intersection; ∪ looks like a U for union.
Example 2: reading set notation
Using the diagram, find n(F ∩ T), n(F ∪ T), n(T′) and n(F ∩ T′).
- n(F ∩ T) = 4.
- n(F ∪ T) = 10 + 4 + 7 = 21.
- n(T′) = 10 + 9 = 19.
- n(F ∩ T′) is football but not tennis: 10.
Example 3: listing elements
ξ = {1, 2, 3, …, 12}. A = multiples of 3, B = even numbers. List A ∩ B and A ∪ B.
- A = {3, 6, 9, 12}, B = {2, 4, 6, 8, 10, 12}.
- A ∩ B = {6, 12}. A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}.
Shading regions
Some questions ask you to shade a region described in set notation. Work it out one piece at a time. For A′ ∩ B, first picture A′ (everything outside A), then keep only the part that is also inside B: that is “B only”. For (A ∪ B)′, shade everything outside both circles. Checking with a small list of example items is a reliable way to confirm your answer.
Probability from a Venn diagram
To find a probability, divide the number in the region you want by the total in the universal set.
Example 4: simple probabilities
A student from the class is chosen at random. Find P(plays both) and P(plays at least one sport).
- P(F ∩ T) = 4/30 = 2/15.
- P(F ∪ T) = 21/30 = 7/10.
Conditional probability (Higher)
“Given that” changes the total. If you know the student plays tennis, you only look inside the tennis circle, so the denominator becomes n(T).
Example 5: given that
A student who plays tennis is chosen at random. Find the probability they also play football.
- Only consider the 11 tennis players. Of these, 4 also play football.
- P(F | T) = 4/11. The formula is P(F | T) = P(F ∩ T)/P(T) = (4/30)/(11/30), which gives the same answer.
Three-set Venn diagrams
With three circles there are eight regions. Use the same rule: start with the centre, where all three overlap, then the regions where exactly two overlap, then each “only” region, and finally the outside.
Example 6: three sets
Of 50 students, 20 study French, 18 Spanish and 15 German. 7 study French and Spanish, 5 French and German, 4 Spanish and German, and 2 study all three. How many study none?
- Centre: 2. Exactly French and Spanish: 7 − 2 = 5. French and German only: 5 − 2 = 3. Spanish and German only: 4 − 2 = 2.
- French only: 20 − 5 − 3 − 2 = 10. Spanish only: 18 − 5 − 2 − 2 = 9. German only: 15 − 3 − 2 − 2 = 8.
- Inside the circles: 2 + 5 + 3 + 2 + 10 + 9 + 8 = 39. None: 50 − 39 = 11.
Example 7: forming an equation
In a group of 40, x people like both tea and coffee. 25 like tea and 22 like coffee, and 5 like neither. Find x.
- Tea only: 25 − x. Coffee only: 22 − x.
- (25 − x) + x + (22 − x) + 5 = 40, so 52 − x = 40.
- x = 12.
Common mistakes
- Not subtracting the overlap, so items in both sets are counted twice.
- Forgetting the region outside all the circles.
- Mixing up ∩ (and) with ∪ (or).
- Using the whole total as the denominator for a “given that” question.
Practice questions
Try each one before opening the answer.
Q1. Of 40 people, 25 have a dog, 18 have a cat and 8 have both. How many have neither?
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Q2. Using the same data, find P(dog only).
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Q3. A = {1, 2, 3, 4}, B = {3, 4, 5}. Find A ∪ B.
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Q4. Using the dog and cat data, find P(cat | dog).
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Frequently asked questions
Are Venn diagrams on the Foundation paper?
Yes. Filling in Venn diagrams and simple probabilities from them are on both tiers. Conditional probability and harder set notation are mainly Higher.
When should I use a Venn diagram instead of a tree diagram?
Use a Venn diagram when items can belong to overlapping groups at the same time. Use a tree diagram for events that happen one after another.
Key takeaways
- Start in the middle and work outwards.
- ∩ means and; ∪ means or; A′ means not A.
- Check that all regions add to the total.
- Probability = region ÷ total.
- For “given that”, divide by the size of the given set.