Proofs A.Miller Mar 97
Don't
Don't write false statements in a proof (at least in the final form).
(exception: proofs by contradiction)
Don't start a proof by assuming what you are asked to prove. That is, if
the statement is "If P then Q" do not start by saying "Assume Q."
(exception: one style used is to start by saying "We must show Q")
If you know something in your proof is wrong or incomplete, say so. The
best way is to isolate it by stating it as a Lemma or Claim. Say you
think the Claim is true but aren't sure how to prove it, and then use the
statement of the Claim to finish the proof. If you don't do this I will
think that not only are you wrong, but that you don't even know that you
are wrong - which is much worse.
Examples do not prove anything. If they do, we call them counterexamples.
The statements which can be proved are called theorems (or corollaries,
lemmas, claims, propositions). Definitions (or axioms) cannot be proved.
Do
A famous dictum of Polya (a well-known teacher of problem solving skills)
is that "If you can't solve the problem find a simplier version of the
problem, then solve it." Problems can be simplified by considering
special cases or weakening the conclusion or strengthening the hypothesis.
Before starting on a problem make sure you can define anything in the
problem precisely. It is impossible to prove anything about anything
which you cannot define precisely. Also, many proofs consist of almost
nothing but manipulating definitions. Know the definitions so well you
dream about them at night.
You do not have to write or say every detail, but you should be ready to
back up every statement, if challenged. If you write too little, I may
suspect that you don't know why what you wrote is correct. I will never
penalize you for writting too much as long as it is correct.
Be willing to make lots and lots of mistakes. This may sound
ridiculous. However, if a problem is really hard the first 10 things
you try to do will be wrong. The idea is that each "failed idea" gets
closer and closer to what is correct, until eventually you succeed.
Why bother?
Why should you learn proofs? What is a mathematical proof? A
mathematical proof is an organized, clear, correct, logical, persuasive
argument designed to be absolutely airtight and totally convincing. Every
assumption is explicitely laid out and nothing is hidden. It is for this
reason that mathematics is so much simplier than anything else. Writting
good proofs means learning better communication, persuasion and logical
skills. Also, being able to write a proof leads to a deeper understanding
of the mathematics which is neccessary for it to be used properly.