Define the relation R by : $R=\{(x, y) \in W \times W \mid$ the words x and y have at least one letter in common. Then R is-

Question

Define the relation $\mathrm{R}$ by : $R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common. Then $R$ is-

(1) reflexive, symmetric and not transitive

(2) reflexive, symmetric and transitive

(3) reflexive, not symmetric and transitive

(4) not reflexive, symmetric and transitive

Answer 1

SOLUTION : $x, x) \in R \quad \forall x \in W$

$\Rightarrow \quad \mathrm{R}$ is reflexive

Let $\quad(x, y) \in R$, then $(y, x) \in R$

[ $\because x, y$ have at least one letter in common]

$\Rightarrow \quad \mathrm{R}$ is symmetric.

But $R$ is not transitive

eg. (TALL) R (LIGHT) and (LIGHT) R (HIGH) but (TALL) R (HIGH)