The Art and Craft of Problem Solving

40 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS

more on its intended audience than anything else. For example, many professional

mathematicians would accept "Maximize balanced wires!" as a complete and clear

solution to the Affinnative Action problem (Example 2.1. 9 on page 21).

This book is much more concerned with the process of investigation and discovery

than with polished mathematical argument. Nevertheless, a brilliant idea is useless

if it cannot be communicated to anyone else. Furthennore, fluency in mathematical

argument will help you to steer and modify your investigations.^6

At the very least, you should be comfortable with three distinct styles of argument:

straightforward deduction (also known as "direct proof'), argument by contradiction,

and mathematical induction. We shall explore them below, but first, a few brief notes

about style.

Common Abbreviations and Stylistic Conventions

1. Most good mathematical arguments start out with clear statements of the hy­
   pothesis and conclusion. The successful end of the argument is usually marked
   with a symbol. We use the Halmos symbol, but some other choices are the ab­
   breviations

QED for the Latin quod erat demonstrandum ("which was to be demon­

strated") or the English "quite elegantly done";

AWD for "and we're done";

W^5 for "which was what we wanted."

2. Like ordinary exposition, mathematical arguments should be complete sen­
   tences with nouns and verbs. Common mathematical verbs are
   ¥, �, 2, <, >, E, C, =*, �.
   (The last four mean "is an element of," "is a subset of," "implies" and "is
   equivalent to," respectively.)


3. Complicated equations should always be displayed on a single line, and labeled
   if referred to later. For example:

1: e-x

2

dx = Vi, (3)

4. Often, as you explore the penultimate step of an argument (or sub-argument),
   you want to mark this off to your audience clearly. The abbreviations TS and
   ISTS ("to show" and "it is sufficient to show") are particularly useful for this
   purpose.


5. A nice bit of notation, borrowed from computer science and slowly becoming
   more common in mathematics, is ":=" for "is defined to be." For example,
   A := B + C introduces a new variable A and defines it to be the sum of the

already defined variables Band C. Think of the colon as the point of an arrow;

we always distinguish between left and right. The thing on the left side of

(^6) This section is deliberately brief. If you would like a more leisurely treatment of logical argument and
methods of proof, including mathematical induction, we recommend Chapters 0 and 4 .1 of (15).