# why's it necessary to prove a map is onto, for an isomorphism?

Fraleigh defines an isomorphism as:

Let $\langle S, * \rangle$ and $\langle S, * \rangle$ be binary algebraic structures. An isomorphism of $S$ with $S'$ is a one-to-one function $\phi$ mapping $S$ onto $S'$ such that $$\phi(x*y) = \phi(x) *' \phi(y) \text{ for all x, y} \in S$$

The author then describes a recipe for showing that two structures are isomorphic. His third step (after "define $\phi$" and "show $\phi$ is 1-1") is to show that $\phi$ is onto $S'$.

Why is this necessary? It makes sense to me that two structures to need to have the same cardinality for isomorphism, but that's not rigorous. From where in the definition does surjection arise?

• Onto-ness arises in the definition when the author says that $\phi$ maps $S$ onto $S'$.
– user61527
Dec 23 '13 at 4:10
• Wow! I am BLIND. Thank you. Dec 23 '13 at 4:13