Question

We say a function f : A B is surjective (onto) iff for all ye B, EA exists so that f() = y. What does it mean for a function not to be surjective?

Answer 1

We say that a function
`f:A\rightarrow B` is surjective or onto if and only if for every element
`b\in B` there exist an element
`a\in A`, such that
`b=f(a)`.

This definition says that a function is surjective if every element of
`B` has a pre-image, or that the image of
`A` by
`f` ‘‘fills’’
`B` completely.

So, if the function
`f` is not surjective, there is, at least, one element of
`B` without pre-image. In other words, that we a have a proper inclusion
`f(A)\subset B`, and
`f(A)\\eq B` .

Explanation: