Section 5.8 Injections and surjections
The condition to be a bijection is really the conjunction of two conditions: first, that we never use the same label twice, and second, that every label be used. Let's give names to these conditions. An injection is a map \(f\colon S \to T\) such that for any \(t\in T\text{,}\) there is at most one
\(s\in S\) such that \(f(s)=t\text{.}\) A surjection is a map \(f\colon S \to T\) such that for any \(t\in T\text{,}\) there is (at least one) \(s\in S\) such that \(f(s)=t\text{.}\) Of course, a bijection is a map that is both an injection and a surjection.
Here's an important axiom that we will use in a nontrivial way a couple of times in this class. It's called the Axiom of Choice. It says that if you have a surjection \(f \colon S \to T\text{,}\) then there exists a map \(s \colon T \to S\) such that \(f \circ s = \id\text{.}\) Note that we are not saying that \(s \circ f = \id\) as well; that would imply that \(f\) is a bijection, which isn't always true.
The map \(s\) is not usually an inverse to \(f\text{,}\) and we do not usually use the notation \(f^{-1}\) for \(s\text{.}\) Rather it is what we call a section of \(f\text{.}\) Thus the Axiom of Choice says that every surjection has a section.