TITLE: Are injectivity and surjectivity dual?
QUESTION [9 upvotes]: Are injectivity and surjectivity dual in some sense? Their set-theoretic definitions are quite different. In particular, the injectivity is a property of a function's graph, while surjectivity is a relationship between the range of the function and its codomain.

REPLY [7 votes]: There is an important sense in which they are not strictly dual.
Let X is a non-empty set.
Then f: X --> Y is injective if and only if it has a left inverse. The proof does not require the Axiom of Choice.
But it is not true that f:X --->Y is surjective if and only it has a right inverse, unless you invoke the Axiom of Choice. In fact, this is logically equivalent to the Axiom of Choice.