Set Theory & Venn Diagrams

medium importance~1 Q in Tier 124 formulas⚡ 15 shortcuts5 subtopics

Overview Learn (5)Formulas (24)Examples (5)Practice (60)

Sets, subsets and Venn diagrams: two- and three-group survey counting shared by SSC, Railway, Banking and CAT papers. A handful of formulas — union, inclusion–exclusion and max–min bounds — solve nearly every question. Most errors come from misreading regions, not from hard algebra.

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Learn — 5 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

5 exam-level questions worked step by step.

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Practice & test

60 questions — untimed practice or a timed test with analysis.

Track record in the exam

Test difficulty mix (60 questions)

17 easy30 medium13 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

Two groups with a 'neither' block

very common

A survey gives two group sizes, a total and a line like '5 like neither'.

First find the union: total minus neither.

Then overlap $=$ sum of the groups minus the union.

Example

In a class of 70 students, 45 play chess and 40 play cards. If 10 students play neither game, how many play both?

At least one: $70 - 10 = 60$ .

Both: $45 + 40 - 60 = 25$

Learn this in “Two-Set Venn Diagrams” →

Overlap when everyone is covered

very common

Two group sizes and the total are given, and every person belongs to at least one group.

The union is the whole total.

Overlap $=$ sum of the groups $-$ total.

Example

In an office of 125 people, 95 like tea and 80 like coffee. If every person likes at least one drink, how many like both?

Both $= 95 + 80 - 125 = 50$

Learn this in “Two-Set Venn Diagrams” →

Three groups: at least one

very common

Three group sizes with three pair counts and a triple; 'at least one' or 'none' is asked.

Use the three-set union formula: singles minus pairs plus triple.

For 'none', subtract the union from the total.

Example

In a survey, 32 play cricket, 26 play hockey and 20 play football. 12 play cricket and hockey, 9 play hockey and football, 10 play cricket and football, and 4 play all three. How many play at least one game?

$32 + 26 + 20 - (12 + 9 + 10) + 4 = 51$

Learn this in “Three-Set Venn Diagrams” →

Exactly two or exactly one region

common

The words 'exactly two' or 'exactly one' appear with three groups.

Exactly two $=$ pair sum $-\,3 \times$ triple.

Exactly one $=$ singles $-\,2 \times$ pair sum $+\,3 \times$ triple.

Example

In a group, 45 like tea, 38 like coffee and 32 like milk. 14 like tea and coffee, 11 like coffee and milk, 13 like tea and milk, and 4 like all three. How many like exactly two drinks?

Pair sum $= 38$ ; exactly two $= 38 - 3 \times 4 = 26$

Learn this in “Three-Set Venn Diagrams” →

Minimum overlap (pigeonhole bound)

common

'At least how many...' with two or three groups and a fixed total.

Minimum overlap $=$ sum of the sizes minus the total.

Keep a floor of 0.

Example

In a group of 90 people, 60 like tea and 50 like coffee. At least how many like both?

$60 + 50 - 90 = 20$

Learn this in “Maximum–Minimum Region Problems” →

Maximum overlap

common

'At most how many...' or 'maximum possible' with overlapping groups.

Maximum overlap $=$ the smallest group size (nest one set inside the other).

Example

In a class of 75 students, 48 play cricket and 36 play football. What is the maximum number who play both?

$\min(48, 36) = 36$

Learn this in “Maximum–Minimum Region Problems” →

Subset and power-set counting

common

'How many subsets' of a set with $n$ elements, sometimes with 'proper' or 'non-empty'.

Use $2^n$ ; subtract 1 or 2 for the proper and non-empty proper variants.

Example

A set has 8 elements. How many subsets does it have?

$2^8 = 256$

Learn this in “Sets, Subsets & Power Sets” →

De Morgan complement counting

occasional

The ask uses complements, e.g. $n(A' \cap B')$ , or words like 'outside both'.

Convert with De Morgan to $n(U) - n(A \cup B)$ and finish in two lines.

Example

In a universal set of 100 elements, $n(A) = 60$ , $n(B) = 50$ and $n(A \cap B) = 20$ . Find $n(A' \cap B')$ .

Union $= 90$ , so $n(A' \cap B') = 100 - 90 = 10$

Learn this in “Sets, Subsets & Power Sets” →

Percentage survey caselet

common

All figures are percentages of people or families; 'neither', 'both' or 'exactly one' is asked.

Work on a base of 100.

Every set formula carries over with % signs.

Example

In a survey, 60% read paper A and 45% read paper B. If 25% read neither, what percentage read exactly one paper?

Union $= 75\%$ ; both $= 60 + 45 - 75 = 30\%$ ; exactly one $= 75 - 30 = 45\%$

Learn this in “Survey & Caselet Word Problems” →

Reverse-solve the triple intersection

occasional

The union (or 'everyone covered') is given; the singles and pairs pin down the triple.

Put $t$ in the union formula and solve the linear equation.

Example

$n(A \cup B \cup C) = 70$ with $n(A) = 40$ , $n(B) = 35$ , $n(C) = 30$ , $n(A \cap B) = 15$ , $n(B \cap C) = 12$ and $n(A \cap C) = 14$ . Find $n(A \cap B \cap C)$ .

$70 = 105 - 41 + t$ , so $t = 6$

Learn this in “Three-Set Venn Diagrams” →

Your next step

New here? Start with subtopic 1 in Learn. Revision mode? Jump straight to the test and let it tell you what to fix.