First some context (binary relations)
$F \subseteq W \times X \quad G \subseteq X \times Y \quad H \subseteq Y \times Z $
Does the following equality hold
$(G \circ F)^{-1} = F^{-1} \circ G^{-1}$
I haven't been able to find a counterexample, so I think it holds in general. But how would I prove it?
This identity
$G \circ F = F^{-1} \circ G^{-1}$
I have a counterexample for.
And then the following identity
$(H \circ G) \circ F = H \circ (G \circ F)$
I have not found a counterexample so this could hold. But how would I prove it?
Edit: The task says give a good reason, so it doesn't have to be a mathematical proof.