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

01

The picture: two circles in a box

Draw two overlapping circles inside a rectangle. The rectangle is the universal set.

The four regions are: only A, only B, both, and neither. Their counts always add to the total.

Worked: a class of 60 has 35 tea drinkers and 30 coffee drinkers, and 10 drink neither.

At least one drink: $60 - 10 = 50$ .
Both: $35 + 30 - 50 = 15$ .
Only tea: $35 - 15 = 20$ ; only coffee: $30 - 15 = 15$ .
Check: $20 + 15 + 15 + 10 = 60$ .

Rule: only A $= n(A) - n(A \cap B)$ . Get the overlap first; every other region follows.

02

The union formula

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

"Tea or coffee" means at least one of the two. The plain sum counts the overlap twice, so subtract it once.

The formula runs backwards too: $n(A \cap B) = n(A) + n(B) - n(A \cup B)$ .

When every person belongs to at least one group, the union is the whole group size.

Worked: an office of 90 employees has 55 Hindi speakers and 40 English speakers, and everyone speaks at least one. Both $= 55 + 40 - 90 = 5$ .

Example: A class of 70 has 45 chess players and 40 card players, with 10 playing neither. The union is $60$ , so both $= 45 + 40 - 60 = 25$ .

03

The neither region

"Neither" sits outside both circles. Count the union first, then subtract from the total.

\text{neither} = n(U) - n(A \cup B)

Worked: 90 people, 50 like cricket, 45 like football, 25 like both. Union $= 50 + 45 - 25 = 70$ , so neither $= 90 - 70 = 20$ .

Tip: "At least one" and "neither" always add to the total. They are complements.

04

Exactly one

"Exactly one" counts the two wing regions only.

\text{exactly one} = n(A) + n(B) - 2\,n(A \cap B)

Worked: 200 people, 120 read paper A, 90 read B, 45 read both. Exactly one $= 210 - 90 = 120$ .

Watch: "At least one" subtracts the overlap once; "exactly one" subtracts it twice.

05

Reading the words carefully

The wording decides the region.

"like A or B" — usually the union.
"like A or B but not both" — exactly one.
"do not like either" — neither.
"only A" — the wing region.

Underline the deciding words before computing. One word changes the answer.

Compare two asks on the same data: 60 people, 35 speak Hindi, 30 speak English, everyone speaks at least one.

"How many speak both?" gives $35 + 30 - 60 = 5$ .
"How many speak only Hindi?" gives $35 - 5 = 30$ .

Note: Options in the exam usually include the union answer for an exactly-one question. Read twice, then answer.

06

Ratio problems on four regions

When a region is a multiple of another, name the smaller one $x$ .

Worked: a class of 45, everyone plays cricket or badminton, only-cricket is twice only-badminton, and 9 play both.

Let only badminton $= x$ ; only cricket $= 2x$ .
$x + 2x + 9 = 45$ , so $x = 12$ .
Badminton players $= 12 + 9 = 21$ .

Note: The four-region sum is one linear equation. Ratios only fix the coefficients.

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

Find both from totals and neither

How to spot it:

Two group sizes, the total, and a 'neither' line; the overlap is asked.

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

Method

Union $=$ total $-$ neither.
Both $= n(A) + n(B) -$ union.
Check the four regions add to the total.

Why it works:

Removing the neither block first makes the union the sum the formula needs.

Try this

In a group of 120 people, 70 like tea and 65 like coffee. If 15 like neither, how many like both?

Show solution

At least one: $120 - 15 = 105$ .
Both: $70 + 65 - 105 = 30$ .

Answer

30

Type 2very common2 practice Q

Find neither from totals and both

How to spot it:

Group sizes and the overlap are given; 'how many like none' is asked.

\text{neither} = n(U) - n(A \cup B)

Method

Union $= n(A) + n(B) -$ both.
Neither $=$ total $-$ union.
Keep the neither block outside the circles.

Why it works:

The union covers everyone in a circle, so the total minus it is exactly the outside block.

Try this

In a survey of 90 people, 50 like cricket and 45 like football, and 25 like both. How many like neither?

Show solution

Union: $50 + 45 - 25 = 70$ .
Neither: $90 - 70 = 20$ .

Answer

20

Type 3very common2 practice Q

Only-A / only-B regions

How to spot it:

'Only tea', 'only Hindi', 'have a bicycle but not a scooter' — a single wing region is asked.

n(A) - n(A \cap B)

Method

Find the overlap first.
Subtract it from the group size.
State the wing region.

Why it works:

The overlap is the part of the circle shared with the other, so removing it leaves the pure wing.

Try this

In a class of 100 students, 60 take History, 55 take Geography and 20 take neither. How many take only History?

Show solution

Union: $100 - 20 = 80$ .
Both: $60 + 55 - 80 = 35$ .
Only History: $60 - 35 = 25$ .

Answer

25

Type 4common2 practice Q

Exactly one of the two

How to spot it:

'Exactly one', 'but not both', or 'one of the two only' appears in the question.

n(A) + n(B) - 2\,n(A \cap B)

Method

Compute the union.
Subtract the overlap once more.
Or subtract $2 \times$ both from the plain sum.

Why it works:

Exactly one keeps the wings and drops the shared middle completely.

Try this

In a group of 300 people, 180 speak Hindi, 150 speak English and 60 speak both. How many speak exactly one language?

Show solution

Union: $180 + 150 - 60 = 270$ .
Exactly one: $270 - 60 = 210$ .

Answer

210

Type 5common2 practice Q

Total when everyone is covered

How to spot it:

'Every person plays at least one' with the group size asked.

n(U) = n(A \cup B)

Method

Note that neither is 0.
Total $=$ union.
Apply the union formula.

Why it works:

With no outside block, the total and the union are the same number.

Try this

Every student of a class studies at least one of Hindi or English. 18 study Hindi, 14 study English and 6 study both. How many students are in the class?

Show solution

Total $= n(H \cup E)$ .
$18 + 14 - 6 = 26$ .

Answer

26

Type 6occasional2 practice Q

Ratio between the wing regions

How to spot it:

'Only cricket is twice only badminton' — one region is a multiple of another.

Method

Name the smaller wing $x$ .
Write the four-region sum with the multiples of $x$ .
Solve the linear equation.
Add the overlap back for the full group size if asked.

Why it works:

The four regions add to the total, so the ratio turns it into one linear equation.

Try this

In a class of 48 students, every student plays at least one of cricket or badminton. The number playing only cricket is twice the number playing only badminton. If 9 play both, how many play badminton?

Show solution

Let only badminton $= x$ : $x + 2x + 9 = 48$ .
$3x = 39$ , so $x = 13$ .
Badminton $= 13 + 9 = 22$ .

Answer

22

08

Formula sheet

Union

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

At least one of the two groups.

Only A

n(A) - n(A \cap B)

The wing region outside B.

Exactly one

n(A) + n(B) - 2\,n(A \cap B)

Only A plus only B.

Neither

n(U) - n(A \cup B)

Outside both circles.

09

Shortcuts that save time

⚡ Fill the four regions, then read off

Get the overlap first, then peel off each wing. Every ask — both, only, exactly one — becomes a subtraction.

Example

In a class of 100 students, 60 take History, 55 take Geography and 20 take neither. How many take both subjects?

Show solution

Union: $100 - 20 = 80$ .
Both: $60 + 55 - 80 = 35$ .

Answer

35

⚡ Four regions must add to the total

Only A $+$ only B $+$ both $+$ neither $=$ total. Use it as a check, or to find the one missing region.

Example

In a survey of 150 people, 85 like tea, 80 like coffee and 30 like both. How many like neither?

Show solution

Union: $85 + 80 - 30 = 135$ .
Neither: $150 - 135 = 15$ .

Answer

15

⚡ Exactly one subtracts the overlap twice

At least one is the union; exactly one removes the overlap from it. One subtraction apart.

Example

In a group of 100 people, 60 like tea, 55 like coffee and 25 like both. How many like exactly one drink?

Show solution

Union: $60 + 55 - 25 = 90$ .
Exactly one: $90 - 25 = 65$ .

Answer

65

10

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Putting 'neither' inside the circles — neither $= n(U) - n(A \cup B)$ and sits outside both.

Mistake 02

Reporting $n(A) - n(A \cap B)$ as 'both' — that is only A; both is $n(A \cap B)$ itself.

Mistake 03

Using $n(A) + n(B) - n(A \cap B)$ for exactly one — subtract the overlap twice.

Mistake 04

Answering the union when 'exactly one' is asked — at least one and exactly one differ by the overlap.

Mistake 05

Dropping one region in the final check — the four regions must add exactly to the total.

Mistake 06

Forgetting that 'every person likes at least one' fixes the union at the group size.

11

Quick revision

Read this the night before the exam.

Union: $n(A) + n(B) - n(A \cap B)$ .
Only A $= n(A) - n(A \cap B)$ .
Exactly one $= n(A) + n(B) - 2n(A \cap B)$ .
Neither $= n(U) - n(A \cup B)$ .
Four regions add to the total: wings $+$ both $+$ neither $= n(U)$ .
'Every person at least one' means the union is the whole total.

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