JEE Main 2025 Apr 8 Shift 2, Mathematics Q15: Relations

JEE Main2025Apr 8, Shift 2Mathematics

Let $A = {0, 1, 2, 3, 4, 5}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $max {x, y} \in {3, 4}$ . Then among the statements

$(S_{1})$ : The number of elements in $R$ is 18, and

$(S_{2})$ : The relation $R$ is symmetric but neither reflexive nor transitive

A
both are true
B
both are false
C
only $(S_{2})$ is true
D
only $(S_{1})$ is true

Solution

List the ordered pairs $(x, y)$ from $A \times A$ with $max {x, y} \in {3, 4}$ .

Pairs with $max = 3$ : at least one entry is 3 and neither exceeds 3, giving $(0, 3), (3, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3)$ : 7 pairs.

Pairs with $max = 4$ : at least one entry is 4 and neither exceeds 4, giving $(0, 4), (4, 0), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)$ : 9 pairs.

Total: $7 + 9 = 16$ pairs, so $∣ R ∣ = 16$ and $(S_{1})$ (which claims 18) is false.

Symmetry: swapping $x$ and $y$ preserves $max {x, y}$ , so $R$ is symmetric.

Reflexivity fails since e.g. $(0, 0) \in / R$ because $max {0, 0} = 0 \in / {3, 4}$ .

Transitivity fails: $(0, 3) \in R$ and $(3, 1) \in R$ , but $(0, 1) \in / R$ since $max {0, 1} = 1$ .

So $(S_{2})$ is true, and only $(S_{2})$ holds, matching option (3).

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JEE Main 2025 Apr 8 Shift 2, Mathematics Q15: Relations

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