Class JEE Mathematics Sets, Relations, and Functions Q #1067
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4 Marks 2018 JEE Main 2018 (Online) 15th April Morning Slot MCQ SINGLE
Consider the following two binary relations on the set $A = {a, b, c}$:
$R_1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)}$ and
$R_2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}$.
Then:
(A) both $R_1$ and $R_2$ are not symmetric.
(B) $R_1$ is not symmetric but it is transitive.
(C) $R_2$ is symmetric but it is not transitive.
(D) both $R_1$ and $R_2$ are transitive.
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Correct Answer: C
Explanation
Here, both $R_1$ and $R_2$ are symmetric, as for any $(x, y) \in R_1$, we have $(y, x) \in R_1$ and similarly for any $(x, y) \in R_2$, we have $(y, x) \in R_2$.

In $R_1$, $(b, c) \in R_1$, $(c, a) \in R_1$ but $(b, a) \notin R_1$.

Similarly, in $R_2$, $(b, a) \in R_2$, $(a, c) \in R_2$ but $(b, c) \notin R_2$.

Therefore, $R_1$ and $R_2$ are not transitive. Thus, $R_2$ is symmetric but not transitive.

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