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Last
edited 12/26/2008 11:22:00 AM
USEFUL
BEHAVIORS FOR PROOFS
Rewrite
according to the definitions
Definitions are described in detail in the chapter on definitions. Every definition provides a method of proof:
METHOD: To prove that a statement involving a concept is true,
start by rewriting the statement using the definition of the concept.
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.
0 is nonnegative.
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 “
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.
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.
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.
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 ?)
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.
Look ahead to see what you want to get
and figure out something sneaky but legitimate
to do to what you HAVE so that it becomes what you WANT.
See method addiction and walking blindfolded.