Here we shall give a different, and to some readers more natural, "definition" of the expression (f is 1-1 corr).

We shall still use a function to express a correspondence, so we must still find a way to characterize which functions from AB are one-to-one, but we shall do so in a more descriptive way than to stipulate that there is an inverse function.

Consider a function fAB that is supposed to match A and B member-for-member. Every member of A gets paired with some member of B, but there are a couple of things that might go wrong. First, we might miss some member of B; so a one-to-one correspondence must be what we shall call a "surjection":

I.e., every member of B is reachable through f from some member of A.

The other way that a function could fail to match two classes one-to-one is if it paired two different members of one class against just one member of the other. Of course, a function can only pair one value with each argument, but in general could have the same value for different arguments, so this is what must be excluded. We call a function having a unique argument for each value an injection, and define it thus: