A surjection is a function which takes each value in its codomain at some value in its domain. That is, the range (or image) of the function is equal to its codomain. (For every function, the range is a subset of the codomain.) In adjectival form, we say that a function is surjective or onto.

For instance, the function $f: \mathbb Z \to \mathbb Z$ defined by $f(x) = x+1$ is surjective because for every integer, there exists another integer one more than that integer, but the function $f: \mathbb N \to\mathbb N$ defined by $f(x) = x+1$ is not surjective because there exists a natural number which is not one more than any other natural number.

See also

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