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We are ready to begin looking in earnest at proofs. Most proofs are done in a forward manner. First, you state what you want to prove as a claim. Be sure to identify it as a claim. Do not just write it and expect the reader to know that it is your goal. Then you begin to derive facts that you can use. Do not write anything without justification, since that compromises the proof. Work from facts that you know to new facts that you can conclude.
As you go through a forward proof, you will derive facts. One of the most common difficulties that students have is that they forget the facts that they have derived. But that is easy to avoid. As you go, write each key fact in a fact bank, which is just a place for you to collect known facts. At each step of the proof, consult the fact bank to see whether any of those facts are useful.
Our proofs will need to add things to the fact bank as we go, and it is awkward to show the entire new fact bank at each step. So instead we just write the fact bank inside the proof, numbering each important fact for future reference.
Definition. Integer n is even if and only if there exists an integer m so that n = 2m.
The following illustrates a forward proof using a fact bank. It also requires some ideas that are looked at in more detail later. We call the claim a theorem.
| Theorem. The sum of two even integers is always an even integer. | ||||||||
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Proof. To prove a statement about all even integers, it suffices to select two arbitrary even integers x and y. When we say that x and y are arbitrary even integers, we mean that somebody else has chosen them, and the only thing we know is that they are even integers. It is not acceptable to choose them ourselves. We add to the fact bank: (1) x and y are even integers One of the most important things to remember in doing proofs is to use definitions. Going to the definition of an even integer, and using fact (1), we get (2) There exists an integer u so that x = 2u. Looking at the claim, it is clear that we are interested in x+y. Notice that Defining m = u + v, we see that (4) x + y = 2mBut the definition of an even integer goes both ways. Fact (5) is equivalent to stating that x + y is even.
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