🔢 Quant

Set Theory & Venn Diagrams

Two- and three-set union/intersection counting.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

3/5

of recent CAT years

% of Quant📈

~1%

of section questions

Est. hours⏱️

~4h

to master

2021

~1/22

2022

~1/22

2023

2024

~1/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~0.2 per slot (2022: 0.3 · 2023: 0.3 · 2025: 0.3). Three appearances (2022, 2023, 2025), usually fused with statistics/averages rather than as pure Venn counting.

Set Theory

CAT set questions are counting with overlaps. Inclusion–exclusion (or a Venn diagram filled from the inside out) handles essentially all of them.

The formulas

Two sets: $\;|A\cup B|=|A|+|B|-|A\cap B|$ .

Three sets:

$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|.$

"Neither" $=$ Total $-\;|A\cup B|$ .

A worked example

In a class of 40, 25 like tea, 20 like coffee, and 10 like both. How many like neither?

By inclusion–exclusion, $|T\cup C|=25+20-10=35$ . So

$\text{neither}=40-35=\mathbf 5.$

Filling the Venn from the centre out — both $=10$ , tea-only $=25-10=15$ , coffee-only $=20-10=10$ — gives the same picture and answers any follow-up ("only tea", "exactly one") instantly.

🎯PYQ Evidence

Inclusion-exclusion sets up the relation; the constraints push the answer to a boundary. : the all-three count is squeezed between a maximum (it can't exceed the smallest group, 52) and a minimum (forced when the three totals overshoot the 100-student class), and the gap between those bounds is 47. : with the pairwise overlaps all tied to one variable y, the three-set formula gives 150 = 226 − 4y + t so t = 4y − 76; the constraints t ≥ 1 and t ≤ y then bound y, and pushing to the allowed edge maximises "physics but not maths" at 35. Write every overlap through one variable, then let the at-least/at-most limits decide the extreme.

Common traps

Double-counting the overlap. Subtract $|A\cap B|$ once; for three sets, add the triple back.
"Only A" vs "A". "Only tea" excludes the both-region; read the wording exactly.
Forgetting "neither". The universe often exceeds the union — account for the outsiders.

Checklist

Fill the innermost region first, then work outward
Apply two-set or three-set inclusion–exclusion
Separate "only A" from "A" (with the overlap)
Add "neither" to reach the total

Sample Questions

24 practice questions

Medium

All students of Music High School are in the band, the orchestra, or both. 80% are in only one group. There are 119 students in the band. If 50% of students are in the band only, how many students are in the orchestra only?

Easy

30% of major airline companies equip their planes with wireless internet access. 70% of major airlines offer passengers free on-board snacks. What is the greatest possible percentage of major airline companies that offer both wireless internet and free on-board snacks?

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 3

Medium

In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is

CAT 2023 · Slot 1

Hard

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

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