Question

Let R be a relation in the set of integer I defined by aRb iff a & b both are neither even nor odd. Then show that R is symmetric but neither reflexive nor transitive.

Answer 1

Given relation aRb is defined only when, both a & b are not even or odd at a time.

i.e., if a is even & b is odd then aRb is well defined.

(i) Let a and be odd and even then aRb is defined and aRb ⇒ bRa

Hence, R is symmetric.

(ii) Let a be an odd then aRa is not defined and a be an even then also aRa is not defined.

So, R is not reflexive.

(iii) Let a and b be odd and even respectively. Then if R is transitive, then aRb, bRa ⇒ aRa but, then aRa is not defined as a is odd. Hence, R is not transitive.