Venn Diagrams
🔒 Log in to trackWhat a Venn diagram encodes
🔒 Log in to trackA Venn diagram is a picture of groups. Each circle is one group. Where two circles overlap, the members belong to both groups. First learn to read regions, then learn to add them.
What a Venn diagram shows
A circle is a group of things. Everything inside the circle belongs to that group. Where two circles overlap, the things there belong to both groups.
Take two groups: people who like tea and people who like coffee. The picture has four parts.
| Part | Meaning |
|---|---|
| Only tea | Likes tea, does not like coffee |
| Both | Likes tea and coffee |
| Only coffee | Likes coffee, does not like tea |
| Outside | Likes neither |
Rule: Every person sits in exactly one part. No person is counted in two parts.
Reading the region numbers
Numbers are written inside the parts. Add the parts you need and nothing more.
Suppose only-tea is 20, both is 15, only-coffee is 4 and outside is 10. The total is 20 + 15 + 4 + 10 = 49.
- People who like tea: 20 + 15 = 35. Tea includes the "both" part.
- People who like at least one drink: 20 + 15 + 4 = 39.
- People who like tea only: 20.
Watch: "Likes tea" and "likes only tea" are not the same. The first one includes the overlap.
Turning words into regions
Exam questions describe a region in words. You must find the matching part of the picture.
| Words | Region |
|---|---|
| A and B | The overlap of A and B |
| A or B | Both circles together |
| Only A | A, without the overlap |
| Not A | Everything outside circle A |
| Neither A nor B | Outside both circles |
For three sets, the words "A and B but not C" point to one small region. That is the overlap of A and B, with the part inside C removed.
Tip: Draw the circles and shade the region for each option. The option with the same shading is the answer.
How many regions are there
With one circle there are 2 parts: inside and outside. With two circles there are 4 parts. With three circles there are 8 parts.
The pattern is 2 multiplied by itself once for each circle. Three circles give 2 × 2 × 2 = 8 parts. One part is outside all circles, so 7 parts are inside.
Note: A drawing with three overlapping circles has 7 inside parts. Do not count the outside as a circle part.
Words that need care
Some small words change the region. Read them slowly.
- "Neither A nor B" means outside both circles.
- "Exactly one" means only A plus only B. The overlap is left out.
- "At least one" means all the parts inside the circles.
- "Do not like A" means the total minus everyone in A.
Example: 100 people, and 56 like A. Then 100 − 56 = 44 do not like A.
Example: In a group of 80, 30 like A, 25 like B and 10 like both. At least one: 30 + 25 − 10 = 45. Neither: 80 − 45 = 35.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Phrase to set expression
A phrase such as 'A and B but not C'. The options are written with ∩, ∪ and −.
List the parts of the picture the phrase covers.
For each option, list the parts it covers.
Pick the option with exactly the same parts.
Two expressions mean the same only when they cover the same parts.
Which expression shows 'in A and B but not in C'? (a) A ∩ B ∩ C (b) (A ∩ B) − C (c) (A ∪ B) − C (d) A − (B ∪ C)
Show solutionHide solution
The phrase is one part: inside A and B, outside C.
(a) is the centre part, inside C. Reject.
(c) covers 3 parts, including only-A. Reject.
(d) is only A. Reject.
(b) is A and B with C removed. Match.
(b) (A ∩ B) − C
Reading region numbers
A picture with numbers in the parts. It asks for a group total or an 'only' count.
Read which parts the question wants.
Add only those numbers.
Check that the parts you added are not overlapping.
The parts never overlap, so their numbers add cleanly.
In a diagram of tea and coffee drinkers: only tea 20, both 15, only coffee 4, neither 10. How many like at least one drink?
Show solutionHide solution
At least one = only tea + both + only coffee.
20 + 15 + 4 = 39.
The outside 10 is not added.
39
Counting regions
It asks how many parts or regions a diagram with 2 or 3 circles has.
Count the circles n.
Multiply 2 by itself n times.
Subtract 1 if the outside is not counted.
Each member is inside or outside every circle, which gives 2 choices per circle.
How many separate parts (outside included) does a Venn diagram of 3 overlapping circles have?
Show solutionHide solution
n = 3.
2 × 2 × 2 = 8.
Inside parts = 7, outside = 1.
8
Not and at most
The question says 'do not', 'not in', or 'other than' a group.
Find the total.
Find the members of the group.
Subtract the group from the total.
'Not in A' is everything except A.
100 people were asked. 56 like A. How many do not like A?
Show solutionHide solution
Total = 100.
Like A = 56.
Not A = 100 − 56 = 44.
44
Neither, nor and exactly one
The words 'neither ... nor', 'exactly one' or 'only one' appear with two groups.
Find the union: A + B − both.
For neither, subtract the union from the total.
For exactly one, subtract the overlap from the union.
Neither is outside the circles. Exactly one leaves out the overlap.
In a group of 80, 30 like A, 25 like B and 10 like both. How many like neither? How many like exactly one?
Show solutionHide solution
Union = 30 + 25 − 10 = 45.
Neither = 80 − 45 = 35.
Exactly one = 45 − 10 = 35.
Neither 35, exactly one 35
Formula sheet
The overlap is counted twice on the right, so subtract it once.
Only A is A minus the overlap.
n circles make 2 to the power n parts, including the outside.
Shortcuts that save time
Do not keep numbers in your head. Write each one in its region. The answer is then a single sum.
60 students. 35 like tea, 30 like coffee, 10 like both. How many like neither?
Show solutionHide solution
Tea only = 35 − 10 = 25.
Coffee only = 30 − 10 = 20.
Inside = 25 + 10 + 20 = 55.
Neither = 60 − 55 = 5.
5
For a 'which expression' question, shade the phrase first. Then shade each option. Pick the match.
Mistakes to avoid
Where most students lose marks on this subtopic.
Adding 35 (tea) and 30 (coffee) and calling it the number who like a drink.
The overlap is counted twice. Subtract it once: 35 + 30 − 10 = 55.
Reading 'like tea' as 'like only tea'.
'Like tea' includes the overlap. 'Only tea' removes it.
Forgetting the outside region.
People who like neither drink sit outside both circles. Total minus union gives them.
Counting three circles as 7 parts in all.
There are 7 parts inside the circles and 1 outside, so 8 in all.
Shading A ∪ B when the phrase says 'A and B'.
'And' means the overlap only. 'Or' means both circles together.
Quick revision
Read this the night before the exam.
Each circle is a group. The overlap is 'both'. The outside is 'neither'.
Two circles give 4 parts. Three circles give 8 parts, 7 of them inside.
'Likes A' includes the overlap. 'Only A' does not.
'Neither A nor B' means outside both circles.
At least one = total minus neither.
Fill numbers into the regions before you add.
Practice: 12 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 12 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.