# Bijective Function **A [[function]] for which every possible output has exactly one corresponding input.** >[!Example] Bijective Function >A *bijective function*, *bijection*, or a *one-to-one correspondence*, is a function $f:X\to Y$ such that for every $y\in Y$, there is *exactly one* $x\in X$ such that $f(x)=y$. > > This means that $f$ is also both [[Injective Function|injective]] and [[Surjective Function|surjective]]. As with a surjective function, the [[image]] of a bijective function is equal its [[codomain]]. > > [!Visual definitions]+ > > In an arrow diagram, a function $f:X\to Y$ is bijective if and only if each element of codomain $Y$ has *exactly one incoming* arrow. > > > > ![[BijectiveOrNotArrows.svg]] > > *** > > On a Cartesian plane, a function $f:\mathbb{R}\to \mathbb{R}$ is bijective if and only if *every possible horizontal line* intersects the graph at *exactly one* point. > > > > ![[BijectiveOrNotGraph.svg]] If $f:X\to Y$ is a bijective function for some finite sets $X$ and $Y$, then the [[Cardinality|cardinalities]] of $X$ and $Y$ are *equal*. $|X|=|Y|$