Bridge to Abstract Mathematics: Mathematical Proof and Structures

174 METHODS OF MATHEMATICAL PROOF, PART 1 Chapter 5

used the other. Finally, notice the relationship between the result proved
in Example 2 and Exercise 4, Article 1.3.
Many definitions in mathematics are structured in such a way as to
lend themselves readily to "proof by cases" of theorems involving them.
Look for this especially in definitions whose statement involves cases. One
such situation is given in Example 5.

EXAMPLE 5 Recall that the absolute value 1x1 of a real number x is defined

by

Prove that lxyl = 1x1 lyl for real numbers x and y.

Solution In proofs such as this the counting techniques introduced in

Article 1.5 may be of value. If a proof is going to be divided into cases,

according to certain criteria, it may be a nontrivial problem to count,

and list systematically, all the cases [recall in particular Exercise 1 l(c)

of Article 1 S]. In this particular instance, which is identical to counting

the subsets of a two-element set or to listing the rows of a truth table

based on two letters, there are exactly 22 = 4 cases, including:

Case 11: If x 20 and y < 0, then xy 10 and lxyl = -(xy) = x(-y) =

1x1 lyl.

Note that in each of these two cases we are able to use a very brief proof

by transitivity. Notice also that in Case I1 we use the fact that 1x1 = -x

if x 10, since 101 = 0 = -0. The formulation and completion of the

other two cases are left as an exercise [Exercise 12(a)]. 0

Sometimes in a proof the course of the argument leads to a statement

"either p, or p, or... or p," from the given hypotheses, where the state-

ments pi are exhaustive, but may not be mutually exclusive. In such situa-

tions it may be appropriate to divide the argument at that stage into n

cases and to try to derive the desired conclusion within each of those cases.

Cases in point are Exercises 2(d) and 4(a).

Exercises

1. (a) Prove that if A, B, and X are sets with A E X and B E X, then A u B E X.
   (b) Prove that if A and B are sets such that A E B, then A u B = B.


2. (a) Prove that if A, X, and Y are sets with X E Y, then A u X G A u Y.
   (b) Prove or disprove the converse of the statement in (a); that is, if A, X, and
   Y are sets with A u X G A u Y, then X E Y. (If true, this would mean that the
   onlywaywecanhaveAuX~Au YisifX~ Y.)