Definition: A function is surjective (or onto) if for every in the codomain of , there is an in the domain of such that
now what that basically means, is that the range of the function is the same as the codomain of the function.
so we have and . so .
so, since is surjective, is the codomain and at the same time, the range of . now what does, is use the function to take some element and put it in and then use the function to take that element and then move it to . this last part of the function is what makes it surjective, since it is the part that maps to the range (which is the codomain).