In this guide
Venn diagram questions come in two forms. In the logical form, you pick the diagram that best shows how three categories relate — for example, mammals, whales, fish. In the numerical form, you read or calculate numbers from overlapping circles — for example, how many students study only mathematics. Both need a clear picture of what inside, overlapping and separate circles mean.
Three basic relationships
| Relationship | Diagram | Example |
|---|---|---|
| One group is part of another | A circle inside another | Sparrows — Birds |
| Groups partly overlap | Overlapping circles | Doctors — Women |
| Groups have nothing in common | Separate circles | Cats — Fish |
Standard three-group patterns
| Groups | Relationship | Diagram |
|---|---|---|
| Delhi, India, Asia | Each inside the next | Three nested circles |
| Teachers, men, graduates | All three overlap | Three mutually overlapping circles |
| Pen, pencil, stationery | Two separate circles inside a larger one | Two circles inside a third |
| Dogs, animals, plants | Dogs inside animals; plants separate | Nested pair plus a separate circle |
| Women, mothers, doctors | Mothers inside women; doctors overlap both | A nested pair overlapped by a third |
| Men, fathers, brothers | Fathers and brothers are both men, and a man can be both | Two overlapping circles inside a larger one |
Worked example: Choose the diagram for Doctors, Women, Singers.
Some doctors are women; some women are singers; some doctors are singers; and a person can be all three. So the answer is three mutually overlapping circles.
Worked example: Choose the diagram for Earth, Moon, Solar System.
The Earth and the Moon are separate bodies, and both are part of the solar system: two separate circles inside a larger circle.
Numerical Venn diagrams
Two sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Worked example: In a group of 60 people, 35 like tea, 30 like coffee and 12 like both. How many like neither?
Like at least one = 35 + 30 − 12 = 53. Neither = 60 − 53 = 7.
Like only tea = 35 − 12 = 23.
Three sets
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
Worked example: In a class of 100, 50 study Hindi, 40 study English and 30 study Sanskrit. 15 study Hindi and English, 10 study English and Sanskrit, 12 study Hindi and Sanskrit, and 5 study all three. How many study none?
At least one = 50 + 40 + 30 − 15 − 10 − 12 + 5 = 88. None = 12.
Reading "only" regions
For three sets, "only A" = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C).
In the example above, only Hindi = 50 − 15 − 12 + 5 = 28.
Reading a given diagram
Some questions show a diagram with shapes (for example, a triangle for teachers, a circle for women, a square for singers) and numbers in each region, and ask which number represents "women teachers who are not singers". Locate the region that is inside the triangle and circle but outside the square.
Common traps
| Trap | Fix |
|---|---|
| Treating overlap as containment | Ask whether every A is B |
| Forgetting to add back the triple overlap | Use the full formula |
| Confusing "only A" with "A" | Subtract all overlaps |
Practice
- Which diagram represents Rose, Flower, Lotus?
- Which diagram represents Engineers, Players, Women?
- Which diagram represents India, Maharashtra, Pune?
- In a survey of 80 people, 45 read newspaper A, 40 read B and 20 read both. How many read neither?
- Of 50 students, 30 play cricket, 25 play football and 10 play both. How many play only football?
Answers: 1. Two separate circles inside a larger one. 2. Three mutually overlapping circles. 3. Three nested circles. 4. 15. 5. 15.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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