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.