surjection

A surjective function (aka surjection, or onto function) is a function $f$ that is a surjective relation, i.e. such that every element of the codomain is mapped by $f$ to some element in the domain.

For each element $y$ of the function's codomain, there exists at least one element $x$ in the function's domain such that $f(x)=y$.

The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

Let $f:X\to Y$. The function $f$ is said to be surjective provided that the codomain $Y$ is the image of the function's domain $X$:
For every $y$ in $Y$,
there exists at least one $x$ in $X$
such that $f(x)=y$.

Symbolically,

See also: injection, bijection