Proof Points Key ideas when proving mathematical ideas

Proof Points  Be Patient. Finding proofs takes time. If you don’t see how to do it right away, don’t worry. Researchers sometimes work for weeks or even years to find a single proof. (Not very encouraging is it?)  Taken from: Introduction to the Theory of Computation. by M. Sipser. 2006

Proof Points  Come back to it. Look over the statement you want to prove, think about it a bit, leave it, and then return a few minutes or hours later. Let the unconscious, intuitive part of your mind have a chance to work.

Proof Points  Be neat. When you are building your intuition for the statement you are trying to prove, use simple, clear pictures and/or text. You are trying to develop your insight into the statement, and sloppiness gets in the way of insight. Furthermore, when you are writing a solution for another person to read, neatness will help that person understand it.

Proof Points  Be concise. Brevity helps you express high- level ideas without getting lost in details. Good mathematical notation is useful for expressing ideas concisely. But e sure to include enough of your reasoning when writing up a proof so that the reader can easily understand what you are trying to say.

Proof Points  Solve specific cases. If you are trying to prove that some property is true for k>0, first try to prove it for k=1. If you succeed, try it for k=2, and so on until you can understand the more general case. If a special case is hard to prove, try a different special case or perhaps a special case of the special case.

Prof Types

Proof Types  Construction  Build an example (using variables instead of values) that meets all the requirements of the theorem  Works best for theorems that have statements like: “There exists…”  Note: this sometimes takes a lot of intuition. Drawing a picture is helpful. Then write a description using math terminology to describe your picture.

Proof Types  Contradiction  Assume the opposite of what you are trying to prove.  Go through the proof until you reach a contradiction.  Provided your logic is correct, your original assumption must be false, and thus what you are trying to prove is true.  Warning: getting the opposite right can be tough.

Proof Types  Induction  Prove theorem true for the base case (the base case isn’t always the k=1 term)  Assume true for the kth case (include your assumption statement in the proof)  Using the assumption, prove the theorem true for the k+1th case.  Note: induction works incredibly well for proving recursive algorithms.