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

(B) reflexive, symmetric and not transitive

(A) not reflexive, symmetric and transitive

(C) reflexive, symmetric and transitive

(D) reflexive, not symmetric and transitive

JEE Main · 2006 · 30 Apr 2006 · Offline · Sets Relations and Functions

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

Answer: (B)

Check the three properties in turn. A word shares all of its letters with itself, so every pair

(x, x)

lies in

R

and the relation is reflexive. If

x

and

y

have a letter in common then

y

and

x

have that same letter in common, so

R

is symmetric. Transitivity fails: take

x =

CAT,

y =

TOY and

z =

BOY; CAT and TOY share the letter T, and TOY and BOY share O and Y, yet CAT and BOY have no letter in common, so the pair

(x, z)

is missing. Thus

R

is reflexive and symmetric but not transitive.

JEE Main · 2006 · 30 Apr 2006 · Offline · Single correct option