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Last edited 12/26/2008 11:22:00 AM

USEFUL BEHAVIORS FOR PROOFS

Contents

Rewrite according to the definitions. 1

Vary your approach. 2

Name your variables. 2

Keep Records. 2

Look ahead. 2

Rewrite according to the definitions

Definitions are described in detail in the chapter on definitions.  Every definition provides a method of proof:

Example

Let’s look at a baby example:

Definition: For any integer n:

¨    n is positive if n > 0.

¨    n is negative if n < 0.

¨    n is nonnegative if .

This definition gives a precise meaning to the words “positive“, “negative“ and “nonnegative“. Any question about whether a given integer is positive or negative must be answered by checking this definition.

Consequences of the definition

Theorem

0 is nonnegative.

Proof

The definition of “nonnegative” says n is nonnegative if .  So to prove 0 is nonnegative, we must show that .  But that is true because  

Similarly, the statement “0 is positive” is false because  “0 > 0” is false.

Using the method

When you are starting to learn about a part of abstract math, the method of rewriting what you want to prove using the definitions is the first thing you should do when faced with a statement to be proved.   In most cases, that will not be enough to give the proof!  But it is a start, and it may suggest what to do next. 

Read the proof under direct method (proofs by rewriting are often by the direct method).  Also read the detailed proof in the chapter on presentation of proofs.  Notice how often the definition of “divides” is used.  That is typical of proofs in the beginning of a subject.  More advanced proofs will typically not refer to the definitions much but instead draw on many theorems that have already been proved.  

Warning

Whether they  use rewrite by definition (at least in their heads) or not is a sharp dividing line between newbies to abstract math and those with even some successful experiences with it.  Those with more experience do it automatically and even unconsciously, and some of those who read the description I give here will react like: 

But that’s trivial!!  Why did you devote so much space to that??  It’s OBVIOUS.

But as math educators who pay attention to their students know, it is not obvious to some beginners.  Rewriting according to the definition may seem trivial, but in abstract math courses, some beginning students could improve their grades by a whole letter grade if they would systematically use it.

Vary your approach

Name your variables

Keep Records

Look ahead

When performing a calculation to solve a problem, you may look ahead to the form the solution must take to guide the manipulations you need to carry out.

Example

Suppose you have a right triangle with legs a and b and hypoteneuse c.  The Pythagorean Theorem says that

.

Suppose you are asked to derive the trig identity

from the Pythagorean Theorem. 

So this is what you are given: .   Look ahead to what you want to prove:  .

By using the definition of sin and cos, this means you want to prove

You can do that this way:  Take the given equation  and divide through by , getting

But if c is not zero, then , so the proof is finished.   (What happens if ?)

How did I know to divide by ?  Answer:  I looked ahead to see what I needed to prove! 

Pep Talk

Dividing both sides of a correct equation by a nonzero number gives you another correct equation. 

¨  You don’t have to have a specific reason for doing it. 

¨  The new equation doesn’t know how it got there and doesn’t care. 

¨  It is correct and can be used as part of a proof. 

I learned this example from David A. Olson.

See method addiction and walking blindfolded.

References on proofs