nLab
injection

Contents

Definition

A function ff from AA to BB is injective if x=yx = y whenever f(x)=f(y)f(x) = f(y). An injective function is also called one-to-one or an injection; it is the same as a monomorphism in the category of sets.

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

In constructive mathematics, a strongly extensional function between sets equipped with tight apartness relations is called strongly injective if f(x)f(y)f(x) \ne f(y) whenever xyx \ne y (which implies that the function is injective). This is the same as a regular monomorphism in the category of such sets and strongly extensional functions (while any merely injective function, if strongly extensional, is still a monomorphism). Some authors use ‘one-to-one’ for an injective function as defined above and reserve ‘injective’ for the stronger notion.

Revised on August 31, 2012 15:20:09 by Urs Schreiber (89.204.139.6)