Three-Set Venn Diagrams — notes, question types & practice · Set Theory & Venn Diagrams

01

The union formula for three sets

n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)

Say it as: add singles, subtract pairs, add back the triple.

Why: a point in exactly two sets is counted twice by the singles, so subtract one copy. A point in all three is counted three times by the singles and three times removed by the pairs — so add one copy back.

Worked: $n(A) = 30$ , $n(B) = 25$ , $n(C) = 20$ ; pairs $10, 8, 9$ ; triple $4$ .

Union $= 75 - 27 + 4 = 52$ .

Rule: The signs run $+, +, +, -, -, -, +$ in that order. Write them before substituting.

02

The seven regions

Three circles cut the union into seven regions: three "only" wings, three "exactly two" petals and one middle.

Fill the regions in this order:

Start from the middle: the triple count.
Each petal: pair count minus the triple.
Each wing: single minus its two petals minus the triple.

Worked: pairs $10, 8, 9$ , triple $4$ : petals are $6, 4, 5$ .

Tip: Fill all seven regions once and every sub-question becomes a read-off.

03

Exactly two and at least two

Pair sums count the triple region three times — once inside each pair.

\text{exactly two} = (p_{AB} + p_{BC} + p_{CA}) - 3t

\text{at least two} = (p_{AB} + p_{BC} + p_{CA}) - 2t

Worked: pairs $16, 14, 12$ , triple $6$ . Exactly two $= 42 - 18 = 24$ ; at least two $= 42 - 12 = 30$ .

Exactly two and the triple cannot overlap, so at least two is just their sum: $24 + 6 = 30$ .

04

Exactly one

\text{exactly one} = n(A) + n(B) + n(C) - 2(p_{AB} + p_{BC} + p_{CA}) + 3t

Worked: singles $26, 22, 18$ ; pairs $9, 7, 5$ ; triple $2$ .

Exactly one $= 66 - 42 + 6 = 30$ .

Watch: 'At least two' includes the triple; 'exactly two' does not. One word changes the coefficient.

05

Solving backwards

Word problems usually hide the triple. Put $t$ in the formula and solve.

Worked: union $70$ ; singles $42, 36, 30$ ; pairs $18, 14, 12$ .

$70 = 108 - 44 + t$ , so $t = 6$ .

When "none" is mentioned, first write union $=$ total $-$ none, then solve for $t$ .

Note: "Every person likes at least one" means the union equals the total. That sentence is often the key equation.

06

The sanity check

No region can be negative. Compute the wings; a negative wing means the data is impossible.

Each wing: $n(A) - p_{AB} - p_{CA} + t$ . If this drops below zero, the given numbers cannot all be true.

Options that force a negative region are wrong choices, not answers.

Watch: Before trusting any answer, check all seven regions are non-negative and add to the union.

07

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Union of three sets

How to spot it:

Three singles, three pairs and a triple are given; 'at least one' is asked.

n(A \cup B \cup C) = \Sigma\,n(A) - \Sigma\,n(A \cap B) + n(A \cap B \cap C)

Method

Add the three singles.
Subtract the three pair counts.
Add the triple back.

Why it works:

Each overlap is removed as many times as it was counted in excess.

Try this

For three sets, $n(A) = 25$ , $n(B) = 20$ , $n(C) = 15$ , $n(A \cap B) = 8$ , $n(B \cap C) = 6$ , $n(A \cap C) = 5$ and $n(A \cap B \cap C) = 3$ . Find $n(A \cup B \cup C)$ .

Show solution

Singles: $60$ .
Pairs: $19$ .
$60 - 19 + 3 = 44$ .

Answer

44

Type 2common2 practice Q

Reverse-solve the triple intersection

How to spot it:

The union (or 'everyone covered') is known, and $n(A \cap B \cap C)$ is asked.

t = n(A \cup B \cup C) - \Sigma\,n(A) + \Sigma\,n(A \cap B)

Method

Write the union formula with $t$ .
Move every known count to one side.
Solve for $t$ .

Why it works:

The union equation is linear in $t$ , so one known union pins it down.

Try this

$n(A \cup B \cup C) = 55$ with $n(A) = 32$ , $n(B) = 28$ , $n(C) = 24$ , $n(A \cap B) = 12$ , $n(B \cap C) = 9$ and $n(A \cap C) = 10$ . Find $n(A \cap B \cap C)$ .

Show solution

Singles: $84$ ; pairs: $31$ .
$55 = 84 - 31 + t$ .
$t = 2$ .

Answer

2

Type 3common2 practice Q

Exactly two regions

How to spot it:

'Exactly two' with three groups — the petals of the diagram are asked.

\Sigma\,n(A \cap B) - 3t

Method

Add the three pair counts.
Subtract $3 \times$ the triple.
Read off the answer.

Why it works:

Each pair sum includes the triple once, so stripping the triple needs three subtractions.

Try this

In a group, 45 like tea, 40 like coffee and 35 like milk. 18 like tea and coffee, 12 like coffee and milk, 15 like tea and milk, and 5 like all three. How many like exactly two drinks?

Show solution

Pair sum: $18 + 12 + 15 = 45$ .
Exactly two: $45 - 3 \times 5 = 30$ .

Answer

30

Type 4common2 practice Q

Exactly one region

How to spot it:

'Exactly one' with three groups — the wings of the diagram are asked.

\Sigma\,n(A) - 2\,\Sigma\,n(A \cap B) + 3t

Method

Add the three singles.
Subtract twice the pair sum.
Add three times the triple.

Why it works:

A wing element is counted once, a petal twice and the middle three times by the singles.

Try this

$n(A) = 35$ , $n(B) = 30$ , $n(C) = 25$ , $n(A \cap B) = 12$ , $n(B \cap C) = 10$ , $n(A \cap C) = 8$ and $n(A \cap B \cap C) = 3$ . How many elements lie in exactly one set?

Show solution

Singles: $90$ ; pairs: $30$ .
$90 - 60 + 9 = 39$ .

Answer

39

Type 5common2 practice Q

At least two regions

How to spot it:

'At least two' or 'two or more' with three groups.

\Sigma\,n(A \cap B) - 2t

Method

Add the three pair counts.
Subtract $2 \times$ the triple.
Or add exactly two and the triple.

Why it works:

The triple is inside every pair, so two of its three copies must be removed.

Try this

In a group, 45 like tea, 40 like coffee and 35 like milk; 20 like tea and coffee, 15 like coffee and milk, 12 like tea and milk, and 5 like all three. How many like at least two drinks?

Show solution

Pair sum: $47$ .
At least two: $47 - 2 \times 5 = 37$ .

Answer

37

Type 6common2 practice Q

None of the three

How to spot it:

'How many play none / like neither of the three' with a fixed total.

\text{none} = n(U) - n(A \cup B \cup C)

Method

Compute the union first.
Subtract it from the total.
Check the union does not exceed the total.

Why it works:

The union covers everyone inside a circle, so the leftover is the outside block.

Try this

In a class of 75 students, 40 play cricket, 35 play hockey and 30 play football. 15 play cricket and hockey, 12 play hockey and football, 10 play cricket and football, and 3 play all three. How many play none?

Show solution

Union: $105 - 37 + 3 = 71$ .
None: $75 - 71 = 4$ .

Answer

4

08

Formula sheet

Union of three sets

n(A \cup B \cup C) = n(A) + n(B) + n(C) - p_{AB} - p_{BC} - p_{CA} + t

$p$ are the pair intersections and $t$ the triple.

Exactly two

p_{AB} + p_{BC} + p_{CA} - 3t

Petals only; the triple is removed from each pair.

At least two

p_{AB} + p_{BC} + p_{CA} - 2t

Exactly two plus the triple.

Exactly one

n(A) + n(B) + n(C) - 2(p_{AB} + p_{BC} + p_{CA}) + 3t

The three wings together.

09

Shortcuts that save time

⚡ Add singles, subtract pairs, add the triple

One line gives the union. Sing the sign pattern $+ + +\ -\ -\ -\ +$ while writing it.

Example

For three sets, $n(A) = 22$ , $n(B) = 18$ , $n(C) = 16$ , $n(A \cap B) = 7$ , $n(B \cap C) = 5$ , $n(A \cap C) = 6$ and $n(A \cap B \cap C) = 2$ . Find $n(A \cup B \cup C)$ .

Show solution

Singles: $56$ ; pairs: $18$ .
$56 - 18 + 2 = 40$ .

Answer

40

⚡ Exactly two: pair sum minus three times the triple

Each pair count hides the triple. Subtract it three times to strip it out of all three pairs.

Example

Pair counts are 16, 12 and 10, and 5 elements lie in all three sets. How many lie in exactly two sets?

Show solution

Pair sum: $38$ .
Exactly two: $38 - 3 \times 5 = 23$ .

Answer

23

⚡ Exactly one: singles minus twice the pairs plus three times the triple

The wings survive only after both double and triple counting is removed.

Example

$n(A) = 24$ , $n(B) = 20$ , $n(C) = 16$ , $n(A \cap B) = 8$ , $n(B \cap C) = 6$ , $n(A \cap C) = 7$ , $n(A \cap B \cap C) = 2$ . How many elements lie in exactly one set?

Show solution

Singles: $60$ ; pairs: $21$ .
$60 - 42 + 6 = 24$ .

Answer

24

10

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Subtracting the triple the wrong number of times — the union formula adds it back once after the pairs are subtracted.

Mistake 02

Using pair sums directly as 'exactly two' — subtract $3t$ for exactly two and $2t$ for at least two.

Mistake 03

Forgetting that 'at least two' includes the all-three region.

Mistake 04

Adding the singles and calling it the union — every overlap must be removed first.

Mistake 05

Leaving a negative region in the diagram — a negative count means the data is inconsistent.

Mistake 06

Mixing 'exactly one' with 'at least one' — they differ by every multi-set region.

11

Quick revision

Read this the night before the exam.

Union: singles $-$ pairs $+$ triple.
Exactly two: pair sum $-\,3t$ .
At least two: pair sum $-\,2t$ .
Exactly one: singles $-\,2 \times$ pairs $+\,3t$ .
None: total $-$ union.
Fill seven regions once; read every answer off.
A negative region means inconsistent data.

12

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 12 min · wrong answers go to your mistake notebook automatically.