Bridge to Abstract Mathematics: Mathematical Proof and Structures

162 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5

EXAMPLE 7 Prove that if M > 0, then the linear function y = f(x) =

Mx + B is increasing on R.

Solution Again, we focus first on the desired conclusion that f is increasing

on R; the hypothesis M > 0 will be employed in the course of the proof.

To prove that f is increasing, we must show that if x, and x, are real

numbers with x, < x,, then f (x,) < f (x,). Hence we begin, or set up, the

proof by letting real numbers x, and x, be given with x, < x,, that is,

we are assuming that x, < x,. Now proving that f(x,) < f (x,) is clearly

the same as proving Mx, + B < Mx, + B. What we have to work with,

as we aim toward this conclusion, is the assumption that x, < x, and the

hypothesis M > 0. Using elementary properties of inequalities, we note

that since x, < x, and M > 0, then Mx, < Mx,. Then since Mx, <

Mx,, we may conclude Mx, + B < Mx, + B, so that f(x,) < f(x,), as

desired.

Explanatory remarks added considerably to the length of the proof in
Example 7. In actual practice, the style of proof you will see in most cir-
cumstances, and should try to write, would go something like this: "To
prove f is increasing on R, assume x, < x,. We must prove f(xl) <
f(x,), that is, Mx, + B < Mx, + B. Sincex, < x, and M > 0, then Mx, <
Mx,. Since Mx, < Mx,, then Mx, + B < Mx, + B, as desired."
Before looking at another proof, let us review the strategy of the proofs
in Examples 6 and 7. In both cases the desired conclusion had essentially
the form (Vx)(p(x) + q(x)). Our first step in both proofs was to let a value
of x be given for which we assume that Ax) is satisfied. This x is general,
or arbitrarily chosen, as opposed to being a specifically identified or named
element, but we fix this x and work with it throughout the remainder of
the proof. Our goal was to show that q(x) is valid, using the given hypoth-
eses-and the assbmption that p(x) is valid. You may have realized already
that this approach is really just the choose or pick-a-point method, intro-
duced in Article 4.1 for proofs of set theoretic inclusion, applied in a more
general setting. A proof of a conclusion with logical form (Vx)Cp(x) -, q(x)),
carried out by using the choose method, is an example of a direct proof.
There are various types of direct proof; we will encounter several in the
remainder of this chapter and in Chapter 6. In Article 6.2 we deal with in-
direct proof.
We return in the next two examples to proofs of set theoretic inclusion,
but for statements of more complicated logical structure than those studied
in Articles 4.1 and 5.1. The theorems proved in Examples 8 and 9 have
as their conclusion a statement that one set is a subset of another. We
noted, in Example 4, that the definition of "subset" has the logical form
(WP(x) -, dx)).

EXAMPLE 8 Prove that if A, X, and Y are any sets with X c Y, then

AnXsAnY.