# Injective Function **A [[function]] for which every possible output has at most one corresponding input.** >[!Example] Injective Function > An *injective* or *one-to-one function*, or an *injection*, is a function $f$ such that > $f(x_{1})=f(x_{2})\implies x_{1}=x_{2}$ > Likewise, > $f(x_{1})\neq f(x_{2})\implies x_{1}\neq x_{2}$ > >[!Visual definitions]+ > > In an arrow diagram, a function $f:X\to Y$ is injective if and only if each element of the [[codomain]] $Y$ has *at most one incoming* arrow. > > > > ![[InjectiveOrNotArrows.svg]] > > *** > > On a Cartesian plane, a function $f:\mathbb{R}\to\mathbb{R}$ is injective if and only if *every possible horizontal line* intersects the graph at *at most one* point. > > > > ![[InjectiveOrNotGraph.svg]] If $f:X\to Y$ is an injective function for some finite sets $X$ and $Y$, then $Y$ has *at least* the same [[cardinality]] as $X$. $|X|\le|Y|$