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Unformatted text preview: MATH 135 Fall 2008 Lecture I Notes Introduction Two of the main purposes of MATH 135 are to teach you about proof (what it means, how to prove things, etc.) and to teach you about precision in mathematics. What is a Proof? A proof is A rigorous mathematical argument which unequivocally demonstrates the truth of a given statement. In other words, in mathematics a proof is a sequence of steps, starting with one or more hypotheses, which follow logically from each other, using the axioms of mathematics, leading to the desired con- clusion. Why Proof? We want to know if a statement is always true. Mathematics is cumulative we use old results to prove new results. If it turned out that one result that we thought was true actually failed sometimes, the results following from it could also be false. What kinds of professions rely on mathematics? Engineering Actuarial science Accounting Weather forecasting Banking . . . Actually, just about every profession relies on math somehow! Would you want to drive on a bridge designed by an engineer who relied mathematics that she was mostly sure of (rather than on mathematics had been proven)? Would you trust an actuary to calculate how much you need to contribute to your pension based on a formula that worked some of the time?...
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