Question

Let $w$ denote the words in the English dictionary .Define the relation $R$ by: $R=\{(x,y)\in W\times W\}$ the words $x$ and $y$ have at least one letter in common. Then $R$ is

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

Answer 1

Answer: (B)

reflexive, symmetric and not transitive

Clearly $(x,x)\in R\forall x\in w$

So, $R$ is reflexive.

Let $(x,y)\in R$ and $(y,x)\in R$ as $x$ and $y$ have atleast on letter in common.

So, $R$ is Symmetric.

But $R$ is not transitive. For example,

Let $x=DELHI,y=DWARKA$ and $z=PARK$

then $(x,y)\in R$ and $(y,z)\in R$ but $(x,z)\notin R$