Class JEE Mathematics Sets, Relations, and Functions Q #998
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4 Marks 2025 JEE Main 2025 (Online) 8th April Evening Shift MCQ SINGLE
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
(S1): The number of elements in $R$ is $18$, and
(S2): The relation $R$ is symmetric but neither reflexive nor transitive
(A) both are false
(B) only (S1) is true
(C) only (S2) is true
(D) both are true
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Correct Answer: C
Explanation
To determine the elements in the relation $R$, we need to find all pairs $(x, y)$ such that $\max(x, y) \in {3, 4}$. The set $A = {0, 1, 2, 3, 4, 5}$.

If $\max(x, y) = 3$, the pairs are $(0, 3), (1, 3), (2, 3), (3, 3), (3, 0), (3, 1), (3, 2)$.
If $\max(x, y) = 4$, the pairs are $(0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (4, 0), (4, 1), (4, 2), (4, 3)$.

Thus, the total number of elements in $R$ is $7 + 9 = 16$. Therefore, statement S1 is false.

For statement S2, we check the properties of $R$:
Reflexivity: $(0, 0)$ is not in $R$, so $R$ is not reflexive.
Symmetry: If $(x, y) \in R$, then $\max(x, y) \in {3, 4}$, which implies $\max(y, x) \in {3, 4}$, so $(y, x) \in R$. Thus, $R$ is symmetric.
Transitivity: $(0, 3) \in R$ and $(3, 1) \in R$, but $(0, 1)$ is not in $R$. Therefore, $R$ is not transitive.

Statement S2 is true since $R$ is symmetric but neither reflexive nor transitive.

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