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Unformatted text preview: WRITING PROOFS Christopher Heil Georgia Institute of Technology A “theorem” is just a statement of fact. A “proof” of the theorem is a logical explanation of why the theorem is true. Many theorems have this form: Theorem I. If statement A is true then statement B is true. This just means that whenever statement A is valid, then statement B must be valid as well. A proof is an explanation of WHY statement B must be true whenever statement A is true. 1. Direct Proofs. There are several ways to write a proof of the theorem “If statement A is true then statement B is true.” We’ll discuss several of them in these pages. It may not be obvious at first which variety of proof to use, but a good rule of thumb is to try a direct proof first. A direct proof. Start by assuming that statement A is true. After all, if statement A is false then there’s nothing to worry about; it doesn’t matter then whether B is true or false. So, suppose that statement A is true—write that down as the first step. This is information that you can use and build on. Now try to proceed logically, one step at a time, building on this information until you have shown that statement B is true. An important point is that a proof is always written in English ! There are mathematical symbols mixed in with the words, but you must write clear, complete, English sentences, one after another until you’ve made your way through to statement B. Finally, write an “end-of-proof” symbol like Q.E.D. or square to show that you’ve finished the proof. Here is an example of a simple theorem and a simple direct proof. Theorem 1. If p is a prime number bigger than 2, then p is odd. Proof. Suppose that p is a prime number and p > 2 . (That’s where we’ve assumed that statement A is true. Now we build on this until we’ve shown that statement B is true.) To show that p is odd, we have to show that p is not divisible by 2 . Now, because p is a prime number, it is divisible only by 1 and itself. Since 2 negationslash = 1 and 2 negationslash = p, the number 2 is not one of the numbers that divides p. Therefore p is not divisible by 2 , and hence p is an odd number. square Admittedly, that was a pretty easy theorem and a pretty easy proof, which I’ve made excessively long just to give you the idea. Most theorems are harder, and you have to sit and think before you get the proof straight. DON’T BE DISAPPOINTED IF YOU DON’T SEE HOW TO DO THE PROOF RIGHT AWAY! Most of the time, it takes THREE sheets of paper to write a proof: c circlecopyrt 2011 by Christopher Heil 1 2 WRITING PROOFS (1) Scratch paper, where you just try out all kinds of ideas, most of which don’t work, until you see something that will work....
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- Summer '07
- Logic, Prime number, Theorem II