Mark the correct alternative in the following question:
Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is
(a) symmetric but not transitive
(b) transitive but not symmetric
(c) neither symmetric nor transitive
(d) both symmetric and transitive
We have,
$R=\{(a, b): a$ is brother of $b\}$
Let $(a, b) \in R$. Then,
$a$ is brother of $b$
but $b$ is not necessary brother of $a$ (As, $b$ can be sister of $a$ )
$\Rightarrow(b, a) \notin R$
So, $R$ is not symmetric
Also,
Let $(a, b) \in R$ and $(b, c) \in R$
$\Rightarrow a$ is brother of $b$ and $b$ is brother of $c$
$\Rightarrow a$ is brother of $c$
$\Rightarrow(a, c) \in R$
So, $R$ is transitive
Hence, the correct alternative is option (b).