Venn Diagrams in Reasoning

What does a Venn diagram show?

Each circle is a group of people or things. Where circles overlap, the people belong to both groups.

Numbers inside the regions tell you how many people are there.

What do the questions ask?

You count people in one region, or in several regions joined together.

Typical asks are only one group, both groups, or at least one group.

Why do students lose marks?

They read a circle as one number. But a circle holds several regions.

The cure is to read the diagram one region at a time.

How many regions does a three-circle diagram have?

Seven regions sit inside the circles. An eighth region lies outside all of them.

Each region has its own number.

What are the seven regions?

Three regions are for only one group. Three are for exactly two groups. One is for all three.

The outside region is for people in none of the groups.

Can you read a circle total?

Use the three-circle figure. Circle A holds 8, 4, 5 and 2.

So A has 8 + 4 + 5 + 2 = 19 people.

Circle B holds 6, 4, 3 and 2, so B has 15. Circle C holds 7, 5, 3 and 2, so C has 17.

Why this works: a circle total includes every region that lies inside it.

How do you read a whole circle?

Add every region inside that circle. That is the total for the group.

A circle total counts the overlaps too.

What does only A mean?

Only A means the part of A that no other circle touches.

It is one single region.

What do both and exactly two mean?

Exactly two means the three overlap strips, without the centre.

Both A and B means the A-B strip plus the centre, if the question allows the third group.

What do at least one and at least two mean?

At least one means every region inside any circle.

At least two means the three strips plus the centre.

Can you read these phrases from the figure?

Only A is 8. Exactly two is 4 + 5 + 3, which is 12.

At least one is 8 + 6 + 7 + 4 + 5 + 3 + 2, which is 35.

All three is the centre, which is 2.

Why this works: each phrase names a set of regions, and you add only those.

What does A but not B mean?

Take all of A. Then remove everything that lies in B.

This leaves the A-only region and the A-C strip.

Example one: only one group

A diagram shows tea 12, coffee 9 and both 5. How many like only tea?

Remove the overlap from the tea circle. The tea circle gives 12 minus 5, which is 7.

Example two: the total

In the same case, how many like tea or coffee?

Add the two circles and subtract the overlap once. That is 12 plus 9 minus 5, which is 16.

Why this works: the 5 who like both sit inside each circle, so adding the circles counts them twice.

Example three: the outside

The class has 25 students. How many like neither drink?

Subtract the 16 who like at least one. That leaves 25 minus 16, which is 9.

Why this works: the circles hold 16 people. Everyone else sits outside them.

Example four: three groups

In the three-circle figure, how many are in A or B?

Add A and B, then take off the A-B overlap once. A has 19 and B has 15. The overlap A-B is 4 + 2 = 6.

So A or B is 19 + 15 – 6, which is 28.

Why this works: the A-B overlap includes its strip and the centre.

How do word problems use them?

They give totals and overlaps, but no picture. You draw the circles yourself.

Start by filling the centre, then work outward.

Why fill the centre first?

The centre is counted inside every other number.

Once you fill it, the strips and the only-regions fall out by subtraction.

Can you see a word problem solved?

In a class of 40, 20 like maths, 15 like science and 6 like both.

Fill the centre first. Both is 6. Maths only is 20 – 6 = 14. Science only is 15 – 6 = 9.

At least one is 14 + 6 + 9 = 29, so neither is 40 – 29 = 11.

Why this works: every number comes from one subtraction off a number you already know.

Can you solve one with three groups?

Of 50 people, 25 like A, 20 like B and 15 like C. Eight like A and B, 6 like B and C, 5 like A and C, and 3 like all three.

At least one is 25 + 20 + 15 – 8 – 6 – 5 + 3, which is 44.

Neither is 50 – 44, which is 6.

Why this works: adding the circles counts the strips twice and the centre three times. Subtracting the strips takes the centre out too often, so we add it back once.

Can you use circles for shape relations?

Yes. One circle inside another means all of the inner group are in the outer group.

This is the same picture you draw for syllogism questions.

What mistakes cost marks?

Counting the overlap twice when you add circle totals.

Forgetting the outside region when the question gives a class total.

How do you stay quick?

Write the region numbers beside the diagram. Tick each one as you add it.

Check that all regions plus the outside add up to the total.

How should you practise?

Solve five Venn questions a day, one of each type.

Then try the quiz below and read every explanation.