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

Answer 1

Answer: (D)

Let $x \in W$. Then $x, x$ have every letter common.
$\therefore (x, x) \in R$. Thus $R$ is reflexive.
Let $(x, y) \in R$. Then $x, y$ have atleast one letter in common.
$\therefore y, x$ have atleast one letter in common. Thus $R$ is symmetric.
Let $x=A N D, y=N E X T, z=H E R$.
Then $(x, y) \in R$ and $(y, z) \in R$. But $(x, z) \notin R$. Thus $R$ is not transitive