Examples of mapping two sets where if the two sets are isomorphic doesn't imply that mapping is also 1 to 1 I am struggling with getting an example of two sets S and T and (onto) mapping f, where the fact S and T are isomorphic does not imply that f is also 1 - 1. If possible could you also give an example in which the fact that they are isomorphic would imply that they are 1 - 1?
Thank You!

Hint: Can you do this for finite $T$ and $S$? What happens if $T$ and $S$ are infinite? Think of shift-like maps  $\mathbb{N}\rightarrow \mathbb{N}$
