Thread: showing that groups are isomorphic

I have to show that the function Ø gives the isomorphism of G with G', by showing that Ø is a one to one function and is onto G'.

From my notes,
(1) if xØ = yØ, then e^x= e^y, so x=y. thus Ø is one to one.
(2) if r in R+, then
(In r)Ø = e ^ (In r) = r
where ( In r ) in R. thus Ø is onto R+.

why does (1) and (2) prove that Ø is one to one and is onto?

thanks!

To prove that a function is one to one, you show that or . Then to prove surjectivity, you want to show that every element in the codomain has a pre-image. For example, suppose and we define it from to . 1 has preimage 1, 4 has preimage 2, and 9 has preimage of 3. So it is surjective.