Venn diagrams and set notation | AQA GCSE Maths Notes

Visual model

Venn diagrams show overlap

A only

A\cap B

B onlyoutside both

\cap\text{ means the overlap}

Middle overlap means intersection.

Everything inside either circle means union.

Check whether the question asks for a probability or a count.

Gold-standard guide

20 mins

What you will learn

Organise overlapping groups and calculate probabilities.

Use a clear step-by-step method for venn diagrams and set notation.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier probability skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Venn diagrams show set membership with circles

Step 1

Fill in the intersection

Both F and S: 8 students

Step 2

Fill in French only and Spanish only

French only: 24 − 8 = 16

Step 3

Find the neither region

16 + 8 + 10 = 34 in at least one language

Watch out

Students forget to subtract the overlap when filling in French only, writing 24 in the French circle without removing the 8 already in the intersection

Union

A union B means in A or B or both.

Intersection

A intersection B means in both A and B.

Worked example

In a class of 40 students, 24 study French (F), 18 study Spanish (S), 8 study both. Find P(studies neither) and P(studies exactly one language).

Fill in the intersection: Both F and S: 8 students.

Fill in French only and Spanish only: French only: 24 − 8 = 16. Spanish only: 18 − 8 = 10.

Find the neither region: 16 + 8 + 10 = 34 in at least one language. Neither = 40 − 34 = 6.

Answer both parts: P(neither) = $\frac{6}{4}0$ = $\frac{3}{2}0$ . P(exactly one) = (16 + 10)/40 = $2\frac{6}{4}0$ = $1\frac{3}{2}0$ .

Final answer

P(neither) = $\frac{3}{2}0$ ; P(exactly one language) = $1\frac{3}{2}0$

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

In a class of 40 students, 24 study French (F), 18 study Spanish (S), 8 study both. Find P(studies neither) and P(studies exactly one language).

4 marks4 minsvenn-diagrams-and-set-notation-worked

Show solution

Worked solution

1.Fill in the intersection: Both F and S: 8 students.
2.Fill in French only and Spanish only: French only: 24 − 8 = 16. Spanish only: 18 − 8 = 10.
3.Find the neither region: 16 + 8 + 10 = 34 in at least one language. Neither = 40 − 34 = 6.
4.Answer both parts: P(neither) = $\frac{6}{4}0$ = $\frac{3}{2}0$ . P(exactly one) = (16 + 10)/40 = $2\frac{6}{4}0$ = $1\frac{3}{2}0$ .

Final answer

P(neither) = $\frac{3}{2}0$ ; P(exactly one language) = $1\frac{3}{2}0$

Mark points

M1: fill in the intersection
M1: fill in french only and spanish only
M1: find the neither region
M1: answer both parts
A1: P(neither) = $\frac{3}{2}0$ ; P(exactly one language) = $1\frac{3}{2}0$

Watch out

Students forget to subtract the overlap when filling in French only, writing 24 in the French circle without removing the 8 already in the intersection.Always place the intersection value first, then calculate the exclusive parts by subtracting.

Diagnosticrecall

n(A) = 15, n(B) = 12, n(A ∩ B) = 5, n(U) = 30. Find n(A ∪ B) and n(neither).

1 mark2 minsvenn-diagrams-and-set-notation-q1

Show solution

Worked solution

1.Spot the skill: Venn diagrams show set membership with circles.
2.Use the fill in the intersection stage first, then fill in french only and spanish only.
3.Keep the final answer visible: n(A ∪ B) = 22; n(neither) = 8.

Final answer

n(A ∪ B) = 22; n(neither) = 8

Mark points

M1: use the correct venn diagrams show set membership with circles. intersection (∩) = both. union (∪) = at least one.n(a ∪ b) = n(a) + n(b) − n(a ∩ b). complement a' = everything not in a.fill the diagram from the inside out: intersection first.
A1: n(A ∪ B) = 22; n(neither) = 8

Watch out

Easyprocedure

P(A) = 0.4, P(B) = 0.35, P(A ∩ B) = 0.15. Find P(A ∪ B).

2 marks3 minsvenn-diagrams-and-set-notation-q2

Show solution

Worked solution

1.Spot the skill: Venn diagrams show set membership with circles.
2.Use the fill in french only and spanish only stage first, then find the neither region.
3.Keep the final answer visible: 0.6.

Final answer

0.6

Mark points

M1: use the correct venn diagrams show set membership with circles. intersection (∩) = both. union (∪) = at least one.n(a ∪ b) = n(a) + n(b) − n(a ∩ b). complement a' = everything not in a.fill the diagram from the inside out: intersection first.
A1: 0.6

Watch out

Mediumreasoning

Set A = {2, 4, 6, 8} and B = {1, 2, 3, 4}. Find A ∩ B and A ∪ B.

3 marks4 minsvenn-diagrams-and-set-notation-q3

Show solution

Worked solution

1.Spot the skill: Venn diagrams show set membership with circles.
2.Use the find the neither region stage first, then answer both parts.
3.Keep the final answer visible: A ∩ B = {2, 4}; A ∪ B = {1, 2, 3, 4, 6, 8}.

Final answer

A ∩ B = {2, 4}; A ∪ B = {1, 2, 3, 4, 6, 8}

Mark points

M1: use the correct venn diagrams show set membership with circles. intersection (∩) = both. union (∪) = at least one.n(a ∪ b) = n(a) + n(b) − n(a ∩ b). complement a' = everything not in a.fill the diagram from the inside out: intersection first.
A1: A ∩ B = {2, 4}; A ∪ B = {1, 2, 3, 4, 6, 8}

Watch out

Hardproblem solving

In a Venn diagram, the 'only A' region has 9, 'only B' has 7, 'both' has 4. Find P(A).

3 marks5 minsvenn-diagrams-and-set-notation-q4

Show solution

Worked solution

1.Spot the skill: Venn diagrams show set membership with circles.
2.Use the answer both parts stage first, then fill in the intersection.
3.Keep the final answer visible: P(A) = $1\frac{3}{2}0$ .

Final answer

P(A) = $1\frac{3}{2}0$

Mark points

M1: use the correct venn diagrams show set membership with circles. intersection (∩) = both. union (∪) = at least one.n(a ∪ b) = n(a) + n(b) − n(a ∩ b). complement a' = everything not in a.fill the diagram from the inside out: intersection first.
A1: P(A) = $1\frac{3}{2}0$

Watch out

Exam-stylemulti-step

A' denotes not A. If P(A) = 0.6, find P(A').

4 marks6 minsvenn-diagrams-and-set-notation-q5

Show solution

Worked solution

1.Spot the skill: Venn diagrams show set membership with circles.
2.Use the fill in the intersection stage first, then fill in french only and spanish only.
3.Keep the final answer visible: 0.4.

Final answer

0.4

Mark points

M1: use the correct venn diagrams show set membership with circles. intersection (∩) = both. union (∪) = at least one.n(a ∪ b) = n(a) + n(b) − n(a ∩ b). complement a' = everything not in a.fill the diagram from the inside out: intersection first.
A1: 0.4

Watch out

Grade 9 stretchproblem solving

In a year group of 120 students, 72 study French, 55 study Spanish and 18 study neither. Find how many study both languages.

4 marks7 minsvenn-g9

Show solution

Worked solution

1.Find the number studying at least one language.
2.Use the union rule and solve for the overlap.

Final answer

25 students

Mark points

M1: union = 102
M1: 72 + 55 - overlap = 102
A1: 25

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Venn diagrams and set notation - 2 marksn(A) = 15, n(B) = 12, n(A ∩ B) = 5, n(U) = 30. Find n(A ∪ B) and n(neither).Mark answer

Answer

n(A ∪ B) = 22; n(neither) = 8

2Probability basics - 2 marksP(A) = 0.35. Find P(not A).Mark answer

Answer

0.65

3Relative frequency - 2 marksExpected frequency of an outcome with probability 0.4 in 250 trials.Mark answer

Answer

100

4Sample spaces and frequency trees - 3 marksP(different colours) when drawing twice with replacement from the bag above.Mark answer

Answer

2 × ( $\frac{4}{6}$ × $\frac{2}{6}$ ) = $\frac{8}{1}8$ = $\frac{4}{9}$

Mastery check

I can explain the method for venn diagrams and set notation.
I can show clear working without skipping key steps.
I can avoid this mistake: Students forget to subtract the overlap when filling in French only, writing 24 in the French circle without removing the 8 already in the intersection.Always place the intersection value first, then calculate the exclusive parts by subtracting.

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