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 $x=AND,y=NEXT,z=HER$.
Then $(x,y)\in R$ and $(y,z)\in R$. But $(x,z)\notin R$. Thus $R$ is not transitive