Let W denote the words in the English dictionary. Define the relation R by :
Let W denote the words in the English dictionary.
Define the relation R by :
R = {(x, y) ∈ W × W | the words x and y have at least one letter in common}.
Then R is
(a) not reflexive, symmetric and transitive
(b) reflexive, symmetric and not transitive
(c) reflexive, symmetric and transitive
(d) reflexive, not symmetric and transitive.
1 Answers
(b) : Given relation R such that R = {(x, y) ∈ W × W | the word x and y have atleast one letter in common}
where W denotes set of words in English dictionary Clearly ( x, x) ∈ R ∀ x ∈ W
∴ (x, x) has every letter common ∴ R is reflexive
Let (x, y) ∈ R then (y, x) ∈ R as x and y have atleast
one letter in common. => R is symmetric.
But R is not transitive
∴ Let x = DON, y = NEST, z = SHE
then (x, y) ∈ R and (y, z) ∈ R. But (x, z) ∉ R.
∴ R is reflexive, symmetric but not transitive.