Question

Let $W$ denote the words in the English dictionary. A relation $R$ defined on $W$ as $xRy$ if the words $x$ and $y$ have atleast one letter common. Then $R$ is

• A. Symmetric and transitive but not reflexive relation.
• B. Reflexive and symmetric but not transitive relation.
• C. Reflexive and transitive but not symmetric relation
• D. Reflexive, symmetric and transitive relation

Answer 1

The correct option is B Reflexive and symmetric but not transitive relation.

$\left(i\right)(x,x)\in R,\forall x\in W,$ So $R$ is reflexive relation.
$\left(ii\right)\text{If}(x,y)\in R,$ then $(y,x)\in R,$
So, $R$ is symmetric relation.
$\left(iii\right)$Let $x=$ CAP, $y=$RAT, $z=$TOY, then $xRy$ (A common) and $yRz$ (T common)
But $xRz$ is not correct as there is no letter common.
So, $R$ is not transitive relation.