Function Warnings Math 323

Math 323 (Laetsch)	Proving Surjective

(Comments similar to those below, following the words “ now for the point ”, apply also to proving that a set T is the image of a set S under a function f.)

As background for the comments below, I'll review the definition of surjective.

Suppose we have a function f : S → T. One definition of surjective in this situation is to say that the range of f equals T. This definition, however, may lead you to believe that you have to prove something TWO ways, as you usually do when you are proving that two sets are equal (if you know how to prove that two sets are equal).
A better way of looking at surjective is to take into account the fact that the notation f : S → T (if used correctly!) implies that T is an appropriate codomain for f, so that the range of f is already known to be a subset of T. (Of course, you have to assume that the person using the notation f : S → T is smart enough to have checked that T really is an appropriate codomain; cf. Defining Functions and Codomains.) Since we already know that the range of f is a subset of the codomain T, all we have to verify for “surjective” is that the codomain T is a subset of the range of f; i.e., every element of T is an element of the range of f. So f : S → T is surjective if and only if —
(*) For every y in T there exists x in S such that f(x) = y
(so that T is a subset of the range of f).

Now for the point:
Many students, when trying to prove that f : S → T is surjective, will start the proof with a statement something like this:

Consider f(x) in T.

This is egregiously wrong, and it is egregiously wrong for TWO reasons --

FIRST, you are trying to prove the statement (*) given above,
"For every y in T, there exists ... ".
To prove this, you should start with an arbitrary
(ARBITRARY ARBITRARY ARBITRARY arbitrary arbitrary arbitrary)
element y of T, not a special element of the form f(x).
SECOND, by starting with f(x) in T, you are essentially assuming what you're trying to prove! You want to start with y in T and prove that there is an x in S such that y = f(x). If you start with something that already is an f(x), there is not much left to prove! Also, don't forget that when you start this correctly, with an arbitrary y in T, and you have found an x, you have to check that x is in the domain of f, not merely that when you plug x into your formula for f, you get y.

(Comments similar to those above apply to proving that a set T is the image of a set S under a function f.)

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Last modified 3/27/08