(i) If [itex](X,*) [/itex] is a binary operation, show that the identity function
[itex] Id_X : X \rightarrow X [/itex]is an isomorphism.
(ii) Let [itex](X_1, *_1) and (X_2, *_2)[/itex] be two binary structures and let [itex]f : X_1 \rightarrow X_2[/itex] be an isomorphism of the binary structures. Show that [itex]f^-1 : X_2 \rightarrow X_1 [/itex] is also an isomorphism.
(iii) Let [itex] (X_1, *_1), (X_2, *_2), (X_3, *_3) [/itex] be three binary structures and
let [itex] f : X_1 \rightarrow X_2 [/itex] and [itex] g : X_2 \rightarrow X_3 [/itex] be isomorphisms of the binary structures. Show that [itex] g \circ f : X_1 \rightarrow X_3 [/itex] is also an isomorphism.
(iv) Denote the statement that [itex](X_1,*_1) [/itex] and [itex] (X_2, *_2) [/itex]are isomorphic by [itex](X_1, *_1) \cong (X_2, *_2)[/itex]. Using the above, show that [itex] \cong[/itex] is reflexive, symmetric and transitive.
The Attempt at a Solution
Okay, so I'm a bit confused with how to work with isomorphisms and binary operations in general. I'm don't know how to approach the first half of the problem, so I can't really do the rest either. Am I supposed to choose elements from the set X and work my proofs from there, or is there some other approach I should be taking? Besides that, I'm not sure I entirely understand the more general premise of the problems:
Starting with i) I'm not sure why the identity function is only isomorphic when there exists a binary relation. I'm not very confident in my understanding, but it seems like the identity function would always be isomorphic?
ii) I'm not sure how to start a proof for this, but since f is an isomorphism, isn't it necessarily bijective so obviously it would have an inverse? I'm confused as to what the proof is supposed to prove?