Given functions $$f_1 : A\to B$$and$$f_2 : A\to B,$$ let us write $f_1 \equiv f_2$ when there exist bijections $\alpha : A\to A$ and $\beta : B \to B$ such that $f_2(\alpha(a)) = \beta(f_1(a))$ for all $a\in A_1$.
(a) Show that $\equiv$ is an equivalence relation.
(b)How many equivalence classes of functions $f : A\to B$ such that $|A| = |B| = 3$ are there?
I tried to prove reflexivity, symmetry and transitivity.
Hints:
Proving reflexivity is very easy. Try using the simplest bijections $\alpha$ and $\beta$ you can think of.
For symmetry, try playing around with $\alpha^{-1}$ and $\beta^{-1}$.
For transitivity, write down what $f_1\equiv f_2$ and $f_2\equiv f_3$ means.