nLab
surjection

Contents

Definition

A function ff from AA to BB is surjective if, given any element yy of BB, y=f(x)y = f(x) for some xx. A surjective function is also called onto or a surjection; it is the same as an epimorphism in the category of sets.

A bijection is a function that is both surjective and injective.

The axiom of choice states precisely that every surjection in the category of sets has a section. Some authors who doubt the axiom of choice use the term ‘onto’ for a surjection as defined above and reserve ‘surjective’ for the stronger notion of a function with a section (a split epimorphism).

Revised on February 5, 2013 22:42:02 by Urs Schreiber (89.204.137.128)