Let W denote the words in the English dictionary. Let the relation R b

1.	Let W denote the words in the English dictionary. Let the relation R be defined by R = {(x, y) ∈ W × W : the words x and y have at least one letter in common}. Then R is (a) Reflexive and transitive, not symmetric (b) Reflexive and symmetric and not transitive (c) Symmetric and transitive, not reflexive (d) Reflexive, symmetric and transitive
Answer» (b) Reflexive and symmetric and not transitive • Let x ∈W. (x, x) ∈R, since the words ‘x’ and ‘x’ have all letters in common ⇒ ‘x’ and ‘x’ have at least one letter in common ⇒ R is reflexive. • Let x, y, z ∈W. Then (x, y) ∈R ⇒ ‘x’ and ‘y’ have at least one letter in common ⇒ ‘y’ and ‘x’ have at least one letters common ⇒ (y, x) ∈R ⇒ R is symmetric. • Let x, y, z ∈W Then (x, y) ∈ R and (y, z) ∈R ⇒ ‘x’ and ‘y‘ have at least one letter common and ‘y‘ and ‘z’ have at least one letter common which does not necessarily mean that ‘x’ and ‘z’ have at lest one letter common. ∴ R is not transitive For example, let x = ‘AND’, y = ‘NOT’, z = ‘PET’ x and y have ‘N’ common y and z have ‘T’ common but x and z have no common letter.

Let W denote the words in the English dictionary. Let the relation R be defined by R = {(x, y) ∈ W × W : the words x and y have at least one letter in common}. Then R is (a) Reflexive and transitive, not symmetric (b) Reflexive and symmetric and not transitive (c) Symmetric and transitive, not reflexive (d) Reflexive, symmetric and transitive

Answer»

(b) Reflexive and symmetric and not transitive

• Let x ∈W.

(x, x) ∈R, since the words ‘x’ and ‘x’ have all letters in common

⇒ ‘x’ and ‘x’ have at least one letter in common

⇒ R is reflexive.

• Let x, y, z ∈W.

Then (x, y) ∈R ⇒ ‘x’ and ‘y’ have at least one letter in common

⇒ ‘y’ and ‘x’ have at least one letters common

⇒ (y, x) ∈R

⇒ R is symmetric.

• Let x, y, z ∈W

Then (x, y) ∈ R and (y, z) ∈R

⇒ ‘x’ and ‘y‘ have at least one letter common and

‘y‘ and ‘z’ have at least one letter common

which does not necessarily mean that ‘x’ and ‘z’ have at lest one letter common.

∴ R is not transitive

For example, let x = ‘AND’, y = ‘NOT’, z = ‘PET’

x and y have ‘N’ common

y and z have ‘T’ common

but x and z have no common letter.