JEE Main 2026 Apr 2 Shift 2, Mathematics Q21: Relations

JEE Main2026Apr 2, Shift 2Mathematics

Let $A = {2, 3, 4, 5, 6}$ . Let $R$ be a relation on the set $A \times A$ given by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \leq w$ . Then the number of elements in $R$ is ______.

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Solution

Count divisibility pairs $(x, z)$ with $x, z \in {2, 3, 4, 5, 6}$ and $x ∣ z$ :

$x = 2$ : $z \in {2, 4, 6}$ — $3$ pairs. $x = 3$ : $z \in {3, 6}$ — $2$ pairs. $x = 4$ : $z = 4$ — $1$ . $x = 5$ : $z = 5$ — $1$ . $x = 6$ : $z = 6$ — $1$ .

Total divisibility pairs $= 3 + 2 + 1 + 1 + 1 = 8$ .

Count pairs $(y, w)$ with $y \leq w$ : for each fixed $y$ , $w$ ranges over ${y, y + 1, \dots, 6}$ . Summing: $5 + 4 + 3 + 2 + 1 = 15$ .

By independence, $∣ R ∣ = 8 \times 15 = 120$ .

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JEE Main 2026 Apr 2 Shift 2, Mathematics Q21: Relations

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