🧩 DILR

Set Theory, Venn & Network Diagrams

Region-counting Venn problems and flow/route network sets.

of DILR

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

2/5

of recent CAT years

% of DILR📈

~4%

of section questions

Est. hours⏱️

~6h

to master

2021

~2/20

2022

2023

3/22

2024

2025

Venn & Network Sets

Two DILR flavours that reward a good picture. Venn sets are counting-with-overlaps across 2–3 groups; network/route sets ask you to enumerate paths or flows. In both, draw the diagram and work from the most constrained region outward.

	2021	2022	2023	2024	2025	Avg/slot
Venn / set-overlap DI sets	–	1.7	–	1.3	–	0.6
Network / flow sets	–	–	–	1.7	–	0.3

🎯PYQ Evidence

A second-tier but real CAT pattern. Venn-based DI sets ran in 2022 (5 questions) and 2024 (4); 2024 also brought a 5-question network/flow set — a format many aspirants had never practised. These sets are won at the diagram: the 2024 venn set, like most, hinged on separating "exactly two" from "at least two" regions. Lower frequency than tables, but when they appear they're usually the most formula-friendly set on the paper — strong solvers bank them fast.

Venn: inclusion–exclusion

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

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

Always fill the centre (all three) first, then the pairwise-only regions, then the singles. "None" $=$ Total $-\;|A\cup B\cup C|$ .

A worked example

✏️Worked Example

Of 100 people: 60 read A, 50 read B, 40 read C; 30 read A&B, 20 B&C, 25 A&C, and 10 read all three. How many read none?

By inclusion–exclusion:

$|A\cup B\cup C|=60+50+40-30-20-25+10=85,$

so none $=100-85=\mathbf{15}$ . (Filling the Venn: centre 10; A&B-only $=30-10=20$ ; A&C-only $=25-10=15$ ; B&C-only $=20-10=10$ ; A-only $=60-30-25+10=15$ ; B-only $=50-30-20+10=10$ ; C-only $=40-25-20+10=5$ — totalling 85.)

Network & route sets

Enumerate complete routes — list every path end to end; never trust a greedy "shortest-looking" guess.
For shortest grid paths from corner to corner ( $m\times n$ blocks), the count is $\dbinom{m+n}{m}$ .
Track capacities and one-way edges; they prune impossible routes — in flow sets (like CAT 2024's), what leaves a node must equal what entered it, and that conservation rule is usually the whole solve.

🎯PYQ Evidence

Lock the structure first — overlap cells for Venn, row/column sums for a network grid — then read off the target. : build a separate three-circle Venn for each region (Asia, Europe, ROW), fix the all-three and pairwise cells from the named-country clues (USA the only all-three, China the only Dheeraj-and-Nitesh), and the bar-chart totals then force the exclusive cells you are asked for. : treat the nine intersections as a 3x3 grid and place the six distinct amounts so each road's end-number equals the sum of ATMs on it — R-A totals 22 = 7 + 15 fixes the extremes, and the distance clues finish it, leaving two valid layouts you check against each question. Whether circles or a grid, pin the fixed cells before counting.

Common traps

⚠️CAT Trap

"Exactly two" vs "at least two." The pairwise overlap $|A\cap B|$ includes the people in all three sets. "Exactly two" regions are $|A\cap B| - |A\cap B\cap C|$ , etc. Recent CAT venn sets are built almost entirely on this distinction — fill the centre first and subtract it out of every pairwise figure before answering anything.

Adding raw group sizes without subtracting overlaps.
Greedy routing that misses a longer-but-valid path.

Checklist

Centre-out filling for Venn regions
Apply 2-set / 3-set inclusion–exclusion
Separate exactly-two from at-least-two
Enumerate routes fully; use $\binom{m+n}{m}$ for grid paths
In flow networks, enforce in = out at every node

Sample Questions

8 practice questions

Easy

In a school, 60 students are in the canteen. 25 of them had coffee and 20 had tea. 5 had both coffee and tea. How many students had neither coffee nor tea?

Easy

In a Venn diagram representing Sun, Moon, and Star: Sun is a type of Star; Moon is neither Sun nor Star. Which diagram correctly represents this relationship?

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CAT PYQ Spotlight

Actual CAT questions on this topic

Context

A bar chart provides information about countries visited by Dheeraj, Samantha and Nitesh, in Asia, Europe and the rest of the world (ROW).

Additional facts:

1. 32 countries were visited by at least one of them.

2. USA (in ROW) is the only country visited by all three.

3. China (in Asia) is the only country visited by both Dheeraj and Nitesh, but not Samantha.

4. France (in Europe) is the only country outside Asia visited by both Dheeraj and Samantha, but not Nitesh.

5. Half of the countries visited by both Samantha and Nitesh are in Europe.

CAT 2024 · Slot 1

TITAMedium

How many countries in Asia were visited by at least one of Dheeraj, Samantha and Nitesh?

Your answer

Context

A bar chart provides information about countries visited by Dheeraj, Samantha and Nitesh, in Asia, Europe and the rest of the world (ROW).

Additional facts:

1. 32 countries were visited by at least one of them.

2. USA (in ROW) is the only country visited by all three.

3. China (in Asia) is the only country visited by both Dheeraj and Nitesh, but not Samantha.

4. France (in Europe) is the only country outside Asia visited by both Dheeraj and Samantha, but not Nitesh.

5. Half of the countries visited by both Samantha and Nitesh are in Europe.

CAT 2024 · Slot 1

TITAMedium

How many countries in Europe were visited only by Nitesh?

Your answer

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