Let be sets and let and be functions.

(1) Suppose is an injective map from to itself. Show that is injective.

(2) Suppose now that and . Show that is bijective and .

My attempt:

(1) is 1-1 such that ,

It follows that . Is this proof correct? Do I need to add any more explanation?

(2) I need some help with question two, so I can get started. Btw, the and notation represents the identity.

Thanks in advance,