= Bijection
{wiki}
= Bijective
{synonym}
A <bijection> $f:X\to Y$ is a <function> that is both <injective> and <surjective>: each element of its <codomain> is the output of exactly one element of its <domain of a function>. The two requirements exclude unused targets and merged inputs respectively. The finite example $f(A)=1$, $f(B)=2$, $f(C)=3$ is a <bijection> from $\{A,B,C\}$ to $\{1,2,3\}$, whereas adding an unused target $4$ makes that same rule fail <surjectivity>.
A <bijection> gives an <inverse function> by reversing every pairing. Conversely, a two-sided <inverse function> implies a <bijection>, as the proof under <inverse function> shows. This is useful for transporting structures without losing or identifying elements. It also defines equality of <cardinality>: two <sets> have equal <cardinality> exactly when a <bijection> pairs them. An infinite example is $n\mapsto2n$ from the <integers> to the even <integers>, with inverse $m\mapsto m/2$ on that specified <codomain>. A proper <subset> can therefore have the same <cardinality> as an infinite <set>. https://discrete.openmathbooks.org/preview/sec_intro-functions.html[Levin, functions: injections, surjections and bijections]{ref}.
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