Venn Diagrams and Set Notation: KS3 Maths Guide

A Venn diagram uses overlapping circles inside a rectangle to show how elements are grouped into sets. The overlapping region holds items belonging to both sets simultaneously. Venn diagrams make it easy to spot unions, intersections, and complements — and they are a powerful tool for calculating probabilities.

What is a set and how is it written?

A set is a collection of distinct objects called elements or members. Sets are usually written with capital letters (A, B) and their elements are listed in curly brackets: A = {1, 3, 5, 7, 9}.

The universal set ξ (or U) contains all the elements under consideration — it is drawn as the rectangle in a Venn diagram.

The empty set ∅ contains no elements.

The number of elements in set A is written n(A). If A = {1, 3, 5, 7, 9} then n(A) = 5.

What do the key set notation symbols mean?

Symbol | Name | Meaning | Example
∩ | Intersection | Elements in BOTH A and B | A ∩ B = "A and B"
∪ | Union | Elements in A OR B (or both) | A ∪ B = "A or B"
A' | Complement of A | Elements NOT in A | A' = "not A"
∈ | Is a member of | An element belongs to a set | 3 ∈ A
∉ | Is not a member of | An element does not belong | 4 ∉ A

How do you draw a Venn diagram?

Worked example: In a class of 30 students, 18 study French (F), 12 study Spanish (S), and 7 study both. Draw a Venn diagram.

Step 1 — Find how many are in each region:

Both F and S (overlap): 7

F only (left circle, no overlap): 18 − 7 = 11

S only (right circle, no overlap): 12 − 7 = 5

Neither (outside both circles): 30 − 11 − 7 − 5 = 7

Step 2 — Draw a rectangle labelled ξ. Draw two overlapping circles labelled F and S.

Step 3 — Place the numbers in the correct regions: 11 in F only, 7 in the overlap, 5 in S only, 7 outside both circles.

Check: 11 + 7 + 5 + 7 = 30 ✓ (equals total number of students)

How do you find probabilities using a Venn diagram?

Once a Venn diagram is complete, divide the number of elements in the required region by the total.

Using the French/Spanish example above (30 students):

P(F ∩ S) = 7/30 (probability of studying both)

P(F ∪ S) = (11 + 7 + 5)/30 = 23/30 (probability of studying French or Spanish or both)

P(F') = (5 + 7)/30 = 12/30 = 2/5 (probability of NOT studying French)

P(F only) = 11/30 (French but not Spanish)

How do you sort numbers into a Venn diagram?

Worked example: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = multiples of 2. B = multiples of 3.

A = {2, 4, 6, 8, 10}

B = {3, 6, 9}

A ∩ B = {6} (multiples of both 2 and 3 = multiples of 6)

A only: {2, 4, 8, 10}

B only: {3, 9}

Neither: {1, 5, 7}

Place each element in the correct region. Check: 4 + 1 + 2 + 3 = 10 elements ✓

What mistakes should you avoid?

Putting elements in the wrong region. Elements in A ∩ B go in the overlapping region only — do NOT write them in the A-only or B-only regions as well. Each element appears exactly once.

Forgetting elements outside both circles. Always check that all elements of the universal set are placed somewhere in the diagram.

Confusing union and intersection. Union (∪) means OR — include anyone in either set. Intersection (∩) means AND — include only those in both sets.

Frequently asked questions

What is the difference between ∩ and ∪?

Intersection (∩) is "and" — only elements belonging to BOTH sets. Union (∪) is "or" — elements belonging to either set or both. A helpful memory trick: the intersection symbol ∩ looks like the letter "n" for "aNd"; the union symbol ∪ looks like a cup that holds everything together.

Can Venn diagrams have three circles?

Yes. Three overlapping circles can represent three sets A, B, and C. There are now seven possible regions: A only, B only, C only, A ∩ B only, A ∩ C only, B ∩ C only, and A ∩ B ∩ C (the centre region where all three overlap). Three-circle Venn diagrams appear at KS3 and GCSE.

What does the complement of a set mean?

The complement A' (sometimes written Ac or Ā) contains all elements of the universal set that are NOT in A. If A = {2, 4, 6} and ξ = {1, 2, 3, 4, 5, 6}, then A' = {1, 3, 5}. Note that P(A) + P(A') = 1 always — a useful probability fact.

How are Venn diagrams different from two-way tables?

Both show how elements are distributed across two categories. A two-way table uses rows and columns to display all combinations; a Venn diagram uses overlapping circles. Venn diagrams are more visual and make the set relationships (union, intersection, complement) easier to see, while two-way tables are better for showing totals and conditional counts systematically.

For Socratic KS3 statistics and probability practice, see aitutors.me.