injection

A function $f$ from $A$ to $B$ is **injective** if $x = y$ whenever $f(x) = f(y)$. 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 $f(x) \ne f(y)$ whenever $x \ne y$ (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 November 7, 2017 09:51:26
by Urs Schreiber
(195.37.234.89)