Question

In the set of all positive integers show that the relation '>' is not an equivalence relation.

Answer 1

A relation $R$ is said to be in equivalence relation if it is reflexive, symmetric and transitive.

Let a relation $R$ in $Z^{+}\times Z^{+}$ defined as $x>y\in R$

Now, if $(a,a)\in$ $Z^{+}\times Z^{+}$

$⟹a>a,$ which is not true

So, $R$ is not reflexive

Thus, the relation '$>$' is not an equivalence relation on $Z^{+},$ where $Z^{+}$ is set of all positive integers the set of positive integers.