# Injective, Surjective, Bijective Injections, surjections, and bijections are *class of functions* distinguished by the manner in which *arguments* (/posts/input expressions from the *[domain](Domain-Codomain-Range-Image.md)*) and *[images](Domain-Codomain-Range-Image.md)* (output expressions from the *[domain](Domain-Codomain-Range-Image.md)*) are related or *mapped* to each other.  So, given a function: $f : X \rightarrow Y$ We can can define the following two terms: * **Onto**: If each element of the codomain is mapped by *at least* one element of the domain. * **One-to-one**: If each element of the codomain is mapped by *at most* one element of the domain * Injective: **one-to-one** (Every item in $X$ is mapped to a *unique* item in $Y$, via $f$. I.e. no two items in $X$ are mapped to the same value in $Y$) * Surjective: **onto** (Every item in $Y$ is mapped from $X$, via $f$. I.e. all items in $Y$ have some corresponding item in $X$ via the map $f$, no value in $Y$ is left without a corresponding item in $X$) * Bijective: **one-to-one** and **onto** (Every item in $Y$ is mapped to via a unique value in $X$, via $f$ . I.e. all items in $Y$ have a unique corresponding item in $X$, related via the map $f$. No two items in $X$ are mapped to the same item in $Y$) Another way of viewing this is:  --- References: * [wikipedia](https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection).