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s3_1 - CHAPTER 3 Methods of Proofs 1 Logical Arguments and...

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CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce one statement from others. A theorem is a proposition that can be proved using definitions, axioms, other theorems, and rules of inference. Discussion In most of the mathematics classes that are prerequisites to this course, such as calculus, the main emphasis is on using facts and theorems to solve problems. Theorems were often stated, and you were probably shown a few proofs. But it is very possible you have never been asked to prove a theorem on your own. In this module we introduce the basic structures involved in a mathematical proof. One of our main objectives from here on out is to have you develop skills in recognizing a valid argument and in constructing valid mathematical proofs. When you are first shown a proof that seemed rather complex you may think to yourself “How on earth did someone figure out how to go about it that way?” As we will see in this chapter and the next, a proof must follow certain rules of inference , and there are certain strategies and methods of proof that are best to use for proving certain types of assertions. It is impossible, however, to give an exhaustive list of strategies that will cover all possible situations, and this is what makes mathematics so interesting. Indeed, there are conjectures that mathematicians have spent much of their professional lives trying to prove (or disprove) with little or no success. 1.2. More Terminology. A lemma is a “pre-theorem” or a result which is needed to prove a theorem. A corollary is a “post-theorem” or a result which follows from a theorem (or lemma or another corollary). 56
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1. LOGICAL ARGUMENTS AND FORMAL PROOFS 57 Discussion The terms “lemma” and “corollary” are just names given to theorems that play particular roles in a theory. Most people tend to think of a theorem as the main result, a lemma a smaller result needed to get to the main result, and a corollary as a theorem which follows relatively easily from the main theorem, perhaps as a special case. For example, suppose we have proved the Theorem: “If the product of two integers m and n is even, then either m is even or n is even.” Then we have the Corollary: “If n is an integer and n 2 is even, then n is even.” Notice that the Corollary follows from the Theorem by applying the Theorem to the special case in which m = n . There are no firm rules for the use of this terminology; in practice, what one person may call a lemma another may call a theorem. Any mathematical theory must begin with a collection of undefined terms and axioms that give the properties the undefined terms are assumed to satisfy. This may seem rather arbitrary and capricious, but any mathematical theory you will likely encounter in a serious setting is based on concrete ideas that have been developed and refined to fit into this setting. To justify this necessity, see what happens if you
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