Question

Let A be the non – empty set of children in a family. The relation ‘x is a brother of y’ in A is

• A. reflexive
• B. symmetric
• C. transitive
• D. an equivalence relation

Answer 1

The correct option is C transitive

Let R denotes the relation ‘is brother of ‘. Let X $ϵ$ A. If x is a girl, then we cannot say that x is brother of x.
$\therefore$ (x,x) $\text{/}ϵ$ R $\therefore$ R is not reflexive.
Let (x, y) $ϵ$ R $\therefore$ x is a brother of y.
$\therefore$ y may or may not be a boy.
$\therefore$ we cannot say that (y, x) $ϵ$ R.
$\therefore$ R is not symmetric.
Let (x, y) $ϵ$ and (y, z) $ϵ$ R.
$\therefore$ x is a brother of y and y is a brother of z
$\Rightarrow$ is brother of z $\Rightarrow$ (x, z) $ϵ$ R
$\therefore$ R is transitive.
$\therefore$ The correct answer is (c).