The problem, from Boolos and Jeffrey's Computability and Logic, states the following definitions:
"If a function $f(a)$ (from $A$ to $B$) is defined for every element $a$ of $A$, then it is called total"
"A correspondence between sets $A$ and $B$ is a one-to-one, total function from $A$ onto $B$."
"Two sets are said to be equinumerous if and only if there is a correspondence between $A$ and $B$."
Then, it asks the reader to show that if $A$ is equinumerous with $B$, then $B$ is equinumerous with $A$. So far, the work I've done is to show that, for a function $f : A \rightarrow B$, there exists a function $f^{-1} : B \rightarrow A$ that is onto (which follows from $f$ being total), and one-to-one (which follows from $f$ existing). However, it remains to be shown that $f^{-1}$ is total. This seems to require that $f$ is onto, which doesn't seem to be stated by the problem. I would appreciate any hints on how to show that $f^{-1}$ must be total, given the constraints.