A function $f:X\to Y$ is injective if it has the property that whenever $f\left(x\right)=f\left(y\right)$, 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 $f\left(n\right)=6$ for all $n$ is not injective: since $f\left(1\right)=f\left(2\right)$ but $1\ne 2$, for instance.