In this guide
Venn diagram questions check one idea: how groups relate to each other. One group can sit wholly inside another, the two can partly overlap, or they can be completely separate. Once you can decide that for every pair of groups, the questions become quick.
In SSC CHSL you will meet three kinds: choose the diagram, read a region, and count people who belong to one group, both or neither. The first two are pure reasoning; the third needs one formula.
How it is asked
- Choose the diagram: three words, such as "Kolkata, West Bengal, India", and four diagram options.
- Read a region: a figure of overlapping shapes, each standing for a group, with a number in each region. "Which number shows educated villagers who are not employed?"
- Counting: "In a class of 50, 30 like cricket, 25 like football and 10 like both. How many like neither?"
The three relationships
| Relationship | Picture | Example |
|---|---|---|
| Inside | One circle within another | Mango, Fruit |
| Overlap | Two circles that cross | Teachers, Women |
| Separate | Two circles that do not touch | Dogs, Birds |
For every pair, ask two questions:
- Is every A a B? If yes, A goes inside B.
- Can something be both A and B? If yes, they overlap. If no, they are separate.
Doing this for all three pairs in a three-word question fixes the diagram.
Common three-group patterns
| Groups | Diagram |
|---|---|
| Kolkata, West Bengal, India | Three nested circles |
| Doctors, Men, Fathers | Fathers inside Men; Doctors overlapping both |
| Tea, Coffee, Beverages | Two separate circles inside a larger one |
| Students, Athletes, Singers | Three mutually overlapping circles |
| Potatoes, Vegetables, Fruits | Potatoes inside Vegetables; Fruits separate |
| Hour, Minute, Second | Three separate circles |
The "Doctors, Men, Fathers" pattern is the one that catches people. Every father is a man, so Fathers sits inside Men. A doctor can be a father, a man who is not a father, or a woman, so Doctors must overlap Fathers, overlap the rest of Men, and extend outside Men.
Reading regions in a figure
Suppose a figure has a circle for educated people, a square for villagers and a triangle for employed people, overlapping so that there are seven regions:
| Region | In circle (educated) | In square (villagers) | In triangle (employed) |
|---|---|---|---|
| 1 | Yes | No | No |
| 2 | No | Yes | No |
| 3 | No | No | Yes |
| 4 | Yes | Yes | No |
| 5 | No | Yes | Yes |
| 6 | Yes | No | Yes |
| 7 | Yes | Yes | Yes |
The method:
- Write down each shape's group.
- Mark the groups the description includes (educated, villagers).
- Mark the groups it excludes ("not employed").
- Find the one region that is inside every included shape and outside every excluded one.
Words such as "only", "but not" and "who are not" are exclusions. Missing one is the usual error.
Counting with overlaps
Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
The overlap is subtracted because people in both groups were counted twice. People in neither group = total − n(A ∪ B). People in A only = n(A) − n(A ∩ B).
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).
For three-set questions it is safer to fill the seven regions from the centre outwards: all three first, then each "exactly two" region, then each "only one" region.
Six worked questions
Q1. Which diagram represents Doctors, Men and Fathers? (a) Three nested circles (b) Fathers inside Men, Doctors overlapping both (c) Three separate circles (d) Three mutually overlapping circles
Every father is a man, so Fathers sits inside Men. Doctors can be fathers, other men or women, so Doctors overlaps both. Answer: (b).
Q2. Which diagram represents Hour, Minute and Second? (a) Three nested circles (b) Two circles inside a third (c) Three separate circles (d) Three overlapping circles
An hour contains minutes, but as classes of things no hour is a minute. Nothing belongs to two of these groups. Answer: (c).
Q3. Which diagram represents Kolkata, West Bengal and India? (a) Three nested circles (b) Two separate circles inside a third (c) Three separate circles (d) Two overlapping circles inside a third
Kolkata is in West Bengal, which is in India. Answer: (a).
Q4. In the figure described above, which region shows educated villagers who are not employed? (a) 4 (b) 5 (c) 6 (d) 7
Included: circle and square. Excluded: triangle. Region 4 is the only one marked Yes, Yes, No. Answer: (a) 4.
Q5. In a class of 50, 30 like cricket, 25 like football and 10 like both. How many like neither, and how many like only cricket? (a) 5 and 20 (b) 5 and 30 (c) 15 and 20 (d) 10 and 20
At least one: 30 + 25 − 10 = 45. Neither: 50 − 45 = 5. Only cricket: 30 − 10 = 20. Answer: (a).
Q6. In a survey of 100 people, 50 read newspaper P, 40 read Q and 30 read R. 15 read P and Q, 10 read Q and R, 12 read P and R, and 5 read all three. How many read none? (a) 8 (b) 12 (c) 17 (d) 22
Union: 50 + 40 + 30 − 15 − 10 − 12 + 5 = 88. None: 100 − 88 = 12. Check by regions: P only 28, Q only 20, R only 13, P and Q only 10, Q and R only 5, P and R only 7, all three 5, which sums to 88. Answer: (b) 12.
Common traps
| Trap | Fix |
|---|---|
| Choosing "overlap" when one group is inside another | Ask "Is every A a B?" first |
| Forgetting that a group can extend outside a larger one | Check whether some A can be outside B |
| Missing a "not" in region questions | List included and excluded groups before looking |
| Adding the overlap instead of subtracting it | n(A ∪ B) = n(A) + n(B) − n(A ∩ B) |
| Reading "both" as "only both" in three-set questions | Fill the centre region first |
Practice
For 1 to 4, choose: (a) three nested circles (b) two separate circles inside a third (c) three mutually overlapping circles (d) one circle inside a second, with the third separate.
- Earth, Moon, Planets
- Pen, Pencil, Stationery
- Teachers, Graduates, Women
- Whales, Mammals, Fish
- Using the figure above, which region shows employed villagers who are not educated? (a) 2 (b) 4 (c) 5 (d) 7
- In a group of 70, 40 read Hindi newspapers, 35 read English ones and 15 read both. How many read neither? (a) 5 (b) 10 (c) 15 (d) 20
- In a group, 25 like tea only, 15 like coffee only and 10 like both. How many like tea? (a) 25 (b) 35 (c) 40 (d) 50
- Everyone in a group of 60 likes cricket or hockey or both. 40 like cricket, of whom 15 also like hockey. How many like only hockey? (a) 15 (b) 20 (c) 25 (d) 35
Answers:
- (d). Earth is inside Planets; the Moon is a satellite, separate from both.
- (b). Pens and pencils are separate, and both are stationery.
- (c). A person can be any combination of teacher, graduate and woman.
- (d). Whales are mammals; fish are separate from both.
- (c) 5. Inside square and triangle, outside circle: No, Yes, Yes.
- (b) 10. 40 + 35 − 15 = 60 read at least one; 70 − 60 = 10.
- (b) 35. Tea only 25 plus both 10.
- (b) 20. Everyone outside the cricket group likes only hockey: 60 − 40 = 20.
What to do next
- For every three-word question, write the answer to "Is every A a B?" and "Can something be both?" for all three pairs.
- Practise ten region questions, listing included and excluded groups before you look at the figure.
- Solve five three-set counting questions by filling regions from the centre outwards.
- Revise syllogism, which uses the same circles, and then take a mixed set from the reasoning revision guide.
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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