Glossary - Glossary of Proof Terms Modern Algebra Fall 2011...

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Glossary of Proof Terms Modern Algebra, Fall 2011 The language of mathematical proofs can sometimes be confusing, since some words and phrases have peculiar meanings different from their meanings in ordinary English. Here is a list of some phrases and the meanings they bear in proofs. Assume. See “ suppose .” This is also often used with the phrase without loss of gener- ality . By definition. This phrase is used to explain that a step in your proof is justified by the very definition of one of the words . (See “ definition .”) Every time you use this phrase, you should actually be thinking about a specific definition. By hypothesis. This phrase is used to indicate that something in your proof is true because it’s one of the hypotheses. (See “ hypothesis .”) Every time you use this phrase, you must be referring to something actually in the statement of the problem. By the inductive hypothesis. If you are doing a proof by induction, you first prove that the statement is true for a “base case” of n = 1, and then you prove that if it is true for n - 1, it is also true for n . The “inductive hypothesis” is the assumption “Suppose it is true for n - 1.” By symmetry. Sometimes you need to prove two statements that are exactly identical, except that two variables (say, x and y ) change places. If the hypotheses about x and y are exactly the same, then once you’ve proved the first statement, you could simply rewrite exactly the same proof , just swapping x and y , to prove the second statement. You can save time and ink by proving the first statement and then saying the second statement holds “by symmetry.” Note that saying “by symmetry” is not the same as saying “ without loss of gener- ality .” Cases. One helpful technique is a proof by cases. For example, suppose you want to prove that n 2 - n is even for all integers n . Claim: Let n Z . Then n 2 - n is even. Proof: We factor n 2 - n = n ( n - 1). Case I: n is even. Then 2 n , so 2 n ( n - 1); thus n 2 - n is even. Case II: n is odd. Then n - 1 is even, so 2 n ( n - 1); thus n 2 - n is again even. Thus in either case, n 2 - n is even. In this proof, we split up into cases that cover all the alternatives: either Case I or Case II must hold. Then we complete the proof in each case, showing that no matter what, the claim holds. Splitting a proof into cases is often a very useful technique; nevertheless, a proof without cases (if one exists) is often prettier.
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Proof Terms Glossary Modern Algebra, Fall 2011 Page 2 of 6 Claim. The word “claim” usually means the main statement you are trying to prove, but it can also refer to some useful fact that will help you complete your proof, but which itself needs first to be justified with its own mini-proof. For example, maybe you have some subset H of a group G , and in order to complete your proof you need to know that H is a sub group of G . The way to do this is simply to embed the claim and its proof within your main proof, like this: .
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