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 commona$\}$ Then R is

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

Answer 1

Answer: (C)

Let $w \in W$ then $(w, w) \in \: R \: \therefore \: R$ is reflexive.
Also if $w_1, w_2 \in\,W $ and $ (w_1,w_2) \in\, R,\,$ then $\, (w_2, w_1) \in\, R.\, \therefore \, R $ is symmetric.
Again Let $w_1 = I N K, w_2$ = L I N K,
$w_3 $ = L E T
Then $(w_1, w_2) \in\, R$
[$\because$ I, N are the common elements of $w_1, w_2](w_2,w_3)\in\,R$
[$\because$ L is the common element of $w_2, w_3$]
But $(w_1, w_3) \notin \,R$
[$\because$ there is no common element of $w_1,, w_3$]
$\therefore $ R is not transitive.