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Unformatted text preview: January 23, 2006; rev. 8/29/06; 9/09 Melvin A. Breuer 1 Mathematical proof techniques EE581 Fall 2009 USC Fillin Answer Only Animation January 23, 2006; rev. 8/29/06; 9/09 Melvin A. Breuer 2 Definitions Theorem: A result stating the ____________ of two or more concepts. From the Greek Oeopeiv (theorein) meaning to “ contemplate .” Lemma: a preparatory result, usually used to reduce the length of a proof, or an intermediate result that can be used many places. Corollary: a result that follows so _________ from a theorem that almost no proof in needed. Conjecture: A statement (theorem) that has not yet been proven, and might be wrong. Proof: The application of a sequence of known facts that result in demonstrating that two or more concepts are ___________ . Axiom: A result assumed to be true. For example, what we learned in K12 Previously “proved” or “correct” theorems, lemmas, and corollaries fillin January 23, 2006; rev. 8/29/06; 9/09 Melvin A. Breuer 3 In mathematics, a proof is a demonstration that, given certain axioms , some statement of interest is necessarily true. Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. In the context of proof theory, where purely formal proofs are considered, such not entirely formal demonstrations in mathematics are often called "social proofs". The distinction has led to much examination of current and historical mathematical practice, quasiempiricism in mathematics, and socalled folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language. Regardless of one's attitude to formalism, the result that is proved to be true is a theorem ; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone. Once a theorem is proved, it can be used as the basis to prove further statements. The socalled foundations of mathematics are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today’s focus is more on practice, i.e. acceptable techniques. Mathematical proofs http://en.wikipedia.org/wiki/Mathematical_proof January 23, 2006; rev. 8/29/06; 9/09 Melvin A. Breuer 4 In mathematics we make assertions about a system, whether it be a number system or something more abstract such as a group or linear space. An assertion that is not known to be true or false is called a hypothesis or conjecture . Prior to 1995, a famous conjecture was Fermat's Last Theorem . It stated that for any integer n ≥ 3, there are no positive integer solutions to the equation x n + y n = z n . (Note that for n =2, 3 2 +4 2 = 5 2 .) The process of establishing the truth of an assertion is called a proof...
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