A function is injective if it has the property that whenever $f (x) = f (y)$ , it is the case that $x = y$ . Given an element in the image, it came from applying $f$ to exactly one element of the domain.

This concept is also commonly called being "one-to-one". That can be a little misleading to someone who does not already know the term, however, because many people's natural interpretation of "one-to-one" (without otherwise having learnt the term) is that every element of the domain is matched up in a one-to-one way with every element of the domain, rather than simply with some element of the domain. That is, a rather natural way of interpreting "one-to-one" is as "bijective" rather than "injective".

Examples

The function $N \to N$ (where $N$ is the set of natural numbers) given by $n \mapsto n + 5$ is injective: since $n + 5 = m + 5$ implies $n = m$ . Note that this function is not surjective: there is no natural number $k$ such that $k + 5 = 2$ , for instance, so $2$ is not in the range of the function.

The function $f : N \to N$ given by $f (n) = 6$ for all $n$ is not injective: since $f (1) = f (2)$ but $1 \neq 2$ , for instance.