In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then the number of perso

NCERT Exemplar · Q36

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Concept understanding — Set Operations

The idea in plain words

Think of your two favourite groups of friends — the ones who play cricket and the ones who play football. Some friends are in both groups, some in only one, and some in neither. Set operations are simply the mathematical ways to answer questions like: "Who's in at least one team?" or "Who's only in the cricket team?".

The whole secret? Each operation is just a different way of combining or comparing two collections — like sorting your friends into different buckets.

Why this works

Sets are just labelled buckets that hold distinct items. The universal set $U$ is the "whole world" of things we're talking about — say, all your friends. Then each operation picks out a specific bucket:

Operation	What it asks	Bucket contains
Union ( $A \cup B$ )	In either?	Everything from $A$ or $B$ (or both)
Intersection ( $A \cap B$ )	In both?	Only the overlap
Difference ( $A \setminus B$ )	In $A$ but not $B$ ?	Just the part of $A$ that doesn't touch $B$
Complement ( $A^c$ )	Not in $A$ ?	Everything outside $A$ (inside $U$ )
Symmetric Difference ( $A \triangle B$ )	In exactly one?	The two crescent-shaped parts, excluding the overlap

Step by step

Let's take two concrete sets so you can see each operation in action:

A = \{1, 2, 3\}, \quad B = \{3, 4, 5\}

Step 1: Union — gather everything from both, but don't repeat anything.

A \cup B = \{1, 2, 3, 4, 5\}

Step 2: Intersection — only what's common to both.

A \cap B = \{3\}

Step 3: Difference ( $A$ minus $B$ ) — start with $A$ , remove anything that's also in $B$ .

A \setminus B = \{1, 2\}

Step 4: Complement — needs a universal set. Let $U = \{1, 2, 3, 4, 5\}$ . Then:

A^c = \{4, 5\}

Step 5: Symmetric Difference — combine the two differences:

A \triangle B = (A \setminus B) \cup (B \setminus A) = \{1, 2\} \cup \{4, 5\} = \{1, 2, 4, 5\}

A common slip …

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Use the principle of inclusion-exclusion for sets: total readers = Hindi + English − both. Subtract from total population to get neither. The answer is 290.

The core idea here is set theory — specifically, the inclusion-exclusion principle. When two groups overlap (people who read both Hindi and English), simply adding the individual counts double-counts the overlap. So we subtract it once to get the actual number of people who read at least one language. Then, "neither" is just the complement: everyone else in the town.

Let’s define:

$H$ = set of persons who read Hindi, $|H| = 450$
$E$ = set of persons who read English, $|E| = 300$
$H \cap E$ = those who read both, $|H \cap E| = 200$
Total population $= 840$

We want $|H^c \cap E^c|$ , the number who read neither.

Find the number who read at least one language By inclusion-exclusion:

|H \cup E| = |H| + |E| - |H \cap E|

|H \cup E| = 450 + 300 - 200 = 550

Subtract from the total to get those who read neither

|H^c \cap E^c| = \text{Total} - |H \cup E|

$= 840 - 550 = 290$ …

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Q.In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then the number of persons who read neither is
(A) $210$
(B) $290$
(C) $180$
(D) $260$

Concept understanding — Set Operations

The idea in plain words

Why this works

Step by step

A common slip …

Step 1 — Compute the union $B\cup C$

Step 2 — Compute the difference $A-(B\cup C)$

Q.In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then the number of persons who read neither is (A) 210210210 (B) 290290290 (C) 180180180 (D) 260260260

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Q.In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then the number of persons who read neither is
(A) $210$
(B) $290$
(C) $180$
(D) $260$

Unlock everything free for 14 days