The given relation is not equivalence

Question

A relation $R$ is defined on the set of integers as follows: $$(a,b)\in R\iff a^b=b^a$$

Clearly, it is reflexive and symmetric. But I am unable to give a counter example that it is not transitive.

Thanks.

Answer 1

Assume $$a^b=b^a $$ and $$b^c=c^b $$

then

$$a^c=b^{\frac {a}{b}c} $$

$$c^a=b^{\frac {c}{b}a} $$

which yields to $$a^c=c^a$$