Bridge to Abstract Mathematics: Mathematical Proof and Structures

170 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5

and x, be real numbers such that x, < x,. Then M = M(x, - x,)/(x, - x,) =

[M(x, - x,) + (B - B)]/(x, - x,) = [(Mx, + B) - (Mx, + B)]/(x, - x,) > 0, since

Mx, + B < Mx, + 6, by our assumption that f is increasing.

(c) FACT The subset S = [O,^11 u [2,^31 of R is not an interval.

Start of "Proof" Using the logical negation of the definition of "interval," as stated

in Example 2, it is sufficient to show that there exist real numbers a, b, and c such

thata<b<c,a~Sandc~Sbut ~ES.

(d) THEOREM For any subsets X and Y of a universal set U, if X c Y, then X u

Y = U.

Start of "Proof" Let X and Y be sets and let w be an arbitrarily chosen element of

X. We must prove WE Y....

(e) FACT [O, 21 is not a subset of [I, 31.

Start of "Proof" Let x be an arbitrary element of [O, 21. We must prove x#

[I, 31....

5.3 Proof by Specialization and

Division into Cases

Proofs of theorems whose conclusion has the form (Vx)(p(x) -, q(x)),

such as those contained in the preceding article, can vary greatly accord-

ing to the specific problems encountered in adapting the assumption that

p(x) is valid, possibly together with some given hypotheses, toward the

desired conclusion q(x). In particular, consider these problems:

EXAMPLE 1 Prove that if a subset C of R x R is symmetric with respect

to both the x axis and the origin, then C is symmetric with respect to

the y axis.

EXAMPLE 2 Given sets A, B, and X, prove that if A n X G B n X and

A n X' G B n X', then A G B.

Both these statements are of the type considered in Article 5.2, since the

conclusion of each (i.e., "C is symmetric with respect to the y axis" and

"A is a subset of B) has a definition of logical form (tlx)(p(x) + q(x)).

Thus in each case we should begin the proof by focusing on that con-

clusion and setting up the proof in terms of its definition.

Specifically, in Example 1, we start by assuming that the ordered pair

(x, y) is an element of C; we must prove that (- x, y) E C, using the given

hypotheses. In Example 2 we begin by letting x be an arbitrary element

of A. We must prove x E B. To accomplish this, we will somehow have

to make use of the two given hypotheses, involving a third set X. Before

reading on, think about these two examples. Can you determine how to

complete the proof of one or both of them? Take some time now to try