Question

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

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

Answer 1

The correct option is A not reflexive, symmetric and transitive

Clearly $(x,x)ϵR\forall xϵW$. So R is relexive.
Let $(x,y)ϵRthen(y,x)ϵR$ as x and y have at least one letter in common. So, R is symmetric.
But R is not transitive for example
Let x = INDIA, y = BOMBAY and z = JOKER then (x, y) $ϵ$ R (A is common) and (y, z) $ϵ$ R (O is common) but (x, z) $\notin$ R. (as no letter is common)