Venn Diagrams: How to Draw Them, Symbols and Examples

A Venn diagram shows how sets relate, using overlapping circles drawn inside a rectangle. The rectangle represents the universal set and each circle represents one set. It is taught in Year 7 as part of the sets topic.

The Parts of a Venn Diagram

How to Draw One

Example: S = {1,2,3,4,5,6,7,8}, A = {1,2,3,4}, B = {3,4,5,6}

1. Draw a rectangle and label it S in the corner.
2. Draw two overlapping circles inside it.
3. Fill the intersection first. The members in both A and B are 3 and 4.
4. Fill A only: 1 and 2. Fill B only: 5 and 6.
5. What remains, 7 and 8, goes outside both circles but still inside the rectangle.

Always start with the intersection. Starting from the outside means the values that belong in the overlap get written twice and the diagram is wrong.

The Four Possible Relationships

• Overlapping — some members shared, but not all. The circles cut through each other.
• Disjoint — no shared members at all. The circles sit apart.
• Subset — every member of A is in B. Circle A sits inside circle B.
• Equal — identical members. The circles coincide.

Solving Word Problems

This is the form that appears most often in tests, and one trick makes it always solvable.

Of 40 students, 25 like maths, 20 like sport, and 5 like neither. How many like both?

Step 1 — let the number who like both be x, then fill the diagram from the intersection outwards:

• Maths only = 25 − x
• Both = x
• Sport only = 20 − x
• Neither = 5

Step 2 — add everything and set it equal to the total:

(25 − x) + x + (20 − x) + 5 = 40

50 − x = 40 → x = 10

So 10 students like both. The equation in step two is an ordinary linear equation in one variable — which is exactly why this chapter follows that one.

The Shortcut Formula

The intersection is subtracted once because otherwise the members in both sets would be counted twice.

The Mistakes That Actually Cost Marks

1. Writing the overlap value twice. If 25 like maths and 10 of them also like sport, the "maths only" region holds 15, not 25. This is the number one cause.

2. Forgetting the group in neither set. The 5 in the example above regularly goes missing, and then the total never adds up.

3. Confusing intersection with union. A memory hook: ∩ looks like the n in "and", while ∪ opens upwards like a container holding everything.

4. Filling the diagram from outside in. Always start with the intersection.

5. Thinking the complement means the other circle. The complement of A is everything in the universe that is not A, including whatever sits outside both circles.