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Unformatted text preview: Introductory Number Theory. First week topics and Homework #1 1 On writing mathematics Part of the objective of this course is to get students to write mathematics as mathematics should be written. Ideally, on completing this course you should be able to 1. Write correct proofs. 2. Be able to distinguish between valid and invalid proofs. Of course, nothing is ideal in this imperfect world. We can only hope for approximations. Here are some pointers/facts/ principles. • Mathematics deals a lot with abstract quantities represented by symbols. The first appearance of any symbol has to be its definition. This can be achieved in many ways, a simple one is by just saying what it is. For example, if you want to use the symbol n to denote an integer, you can say “let n be an integer,” or “ n will denote an integer.” • I used the word “let” in giving an example of how to define an object. Mathematics is not magic , and you cannot use “let” to create things that are not there. For example, you cannot “let n be an integer such that n 2 = 3,” because no such integer exists. You can “let n be an integer such that n 2 = 4,” because there are two integers that fit the bill; 2 and − 2. Another frequent use of “let” is as a substitute for “if” in proofs. In this situation, if you are doing a proof by contradiction, you can actually “let” something be an object that isn’t there. For example, if we want to prove that there is no integer whose square is 3, we could start by saying “let n be an integer such that n 2 = 3,” and then derive some nonsense from that, showing no such n can exist. • All symbols in mathematics have their scope ; they are undefined outside of that scope. Symbols used in a textbook have as their scope at most the textbook. But the scope can be much smaller. For example, I can define a set of integers by: E = { n : n = 2 k for some integer k . } The scope of the symbols n and k is within the brackets ( { , } ) defining E . If you want to use the symbol n outside of E you need to define it; so far, it doesn’t exist outside of E . • A theorem is a proposition that requires a proof. A lemma is a theorem considered not as important to the overall theory, or something that will be later superseded by a theorem. There is, however, no strict rule differentiating theorems from lemmas....
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This note was uploaded on 07/13/2011 for the course MAS 3202 taught by Professor Staff during the Summer '11 term at FAU.
 Summer '11
 STAFF

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