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!

The comments and answer so far given seem to ignore the requirement that $f$ be onto. Here’s an example that takes that into account. Take $S=T=\mathbb{N}$, and let $$f:S\to T:n\mapsto \left\lfloor\frac{n}2\right\rfloor.$$ You should be able without too much trouble to show that $f$ us surjective (maps $S$ onto $T$) and is two-to-one everywhere. (E.g., $f(8) = f(9) = 4$.)
