bijective relation

A bijective relation is a relation that is surjective and injective.

As a surjective relation (right-total), every element of the codomain has at least one preimage element:

• for all $y\in Y$ there exists a $x\in X$ such that $(x,y)\in R$.

As an injective relation (left-unique), every element of the codomain of definition is uniquely assigned:

• for all $x_a,x_b\in X$ and all $y\in Y$, if $(x_a,y)\in R$ and $(x_b,y)\in R$, then $x_a=x_b$.

Thus, to be bijective is for every element of the codomain to have exactly one existing preimage element.