Proving an equivalence relation using inverse functions

[The1TL]

Homework Statement

Let f : A → B be a function and let Γ ⊂ B × B be an equivalence relation on B. Prove that the set (f × f)^-1 (Γ) ⊂ A × A (this can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}) is an equivalence relation on A.

Homework Equations

The Attempt at a Solution

Let (f(a),f(a’)) ⊂ Γ. Since f(a) and f(a’) hold an equivalence relation with each other, it follows that a and a’ hold an equivalence relation with each other. Since f(a) and f(a’) are arbitrary elements of Γ, it follows that (fxf)-1Γ ⊂ A x A is an equivalence relation on A.

I'm not sure if thi is the right approach. In particular I am not sure that i can say that f(a) and f(a') holding an equivalence relation means that a and a' hold one too.

 

[The1TL]

Does anybody know if I am correct? I'm not sure if I'm skipping steps.