Question

Let $W$ denote the words in the English dictionary. Let the relation $R$ be defined by $R=\left\{\right(x,y)\in W\times W:thewordsxandyhaveatleastoneletterincommon\}$. Then $R$ is

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

Answer 1

The correct option is B

Reflexive and symmetric and not transitive

Explanation of the Correct Option:

The correct option is B :Reflexive and symmetric and not transitive

Reflexive: Let $x\in W$.
$(x,x)\in R$, since the words $\text{'}x\text{'}$ and $\text{'}x\text{'}$ have all letters in common
$\Rightarrow \text{'}x\text{'}and\text{'}x\text{'}$ have at least one letter in common
$\Rightarrow R$ is reflexive.

Symmetric: Let $x,y,z\in W$.
Then $(x,y)\in R\Rightarrow \text{'}x\text{'}and\text{'}y\text{'}$ have at least one letter in common
$\Rightarrow \text{'}y\text{'}and\text{'}x\text{'}$ have at least one letters common
$\Rightarrow (y,x)\in R$
$\Rightarrow R$ is symmetric.

Transitive: Let $x,y,z\in W$
Then $(x,y)\in Rand(y,z)\in R$
$\Rightarrow \text{'}x\text{'}and\text{'}y\text{'}$ have at least one letter common and
$\text{'}y\text{'}and\text{'}z\text{'}$ have at least one letter common which does not necessarily mean that $\text{'}x\text{'}and\text{'}z\text{'}$have at least one letter common.
$\Rightarrow R$ is not transitive

Hence, the given relation is Reflexive and symmetric but not transitive

Therefore, the correct option is (B)