JEE Main 2023 (Online) 29th January Evening Shift

Explanation Given that $aRb$ if $2a + 3b = 5m$, $m \in I$. (1) $(a, a) \in R$ as $2a + 3a = 5a$, $a \in N$. Hence, R is reflexive. (2) If $(a, b) \in R$ then $2a + 3b = 5m$. Now, $5(a + b) = 5n$. $3a + 2b + 2a + 3b = 5n$ $\therefore 3a + 2b = 5(n - m)$. $\therefore (b, a) \in R$. $\therefore R$ is symmetric. (3) If $(a, b) \in R$ and $(b, c) \in R$ then $2a + 3b = 5m$, $2b + 3c = 5n$. $\Rightarrow 2a + 5b + 3c = 5(m + n)$. $\Rightarrow 2a + 3c = 5(m + n - b)$. $\therefore (a, c) \in R$. $\therefore R$ is transitive. Hence, R is an equivalence relation.