Question

Let $W$ denote the words in the English dictionary. Define the relation $Rby:$
$R=\left\{\right(x,y)\in W\times W$ the words $x$ and $y$ have atleast one letter in common $\}$. Then $R$ is

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

Answer 1

The correct option is A reflexive, symmetric and not transitive

Let $x$ be a word, $x$ have every letter common.
$\therefore (x,x)\in R$. Thus $R$ is reflexive.
Let $(x,y)\in R$ thus $x,y$ have atleast one letter is common. $y,x$ have atleast one letter in common. Thus $R$ is symmetric.
assume $x=AND,y=NEXT,Z=HER$
then $(x,y)\in R$ and $(y,z)\in R$ But $(y,z)\notin R$
Thus $R$ is not transitive.