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Unformatted text preview: Chapter 1 A Short Introduction to Mathematical Logic and Proof In this course, we are going to take a rigorous approach to the main concepts in single variable Differential Calculus. This means that rather than simply asserting mathematical statements as facts, we will attempt whenever possible, to provide proofs of the validity of these statements. To do so, we will begin with a set of notions and statements that we will take as being given. For example, we will assume the basic notions of set theory and the algebraic and arithmetic properties of the natural numbers, the integers, the rational numbers and the real numbers. We will introduce as axioms some of the perhaps less well-known properties of these objects such as the Principle of Mathematical Induction for the natural numbers and the Least Upper Bound Property for the real numbers. It is important to note that while there is some value in rigour for its own sake and that it is even possible for proofs to be fun, our motivation in this course for including rigour is the hope that we will gain a deeper understanding of the fundamental concepts of Calculus as well as an appreciation for their limitations. In this respect, we will begin with a very brief, and admittedly incomplete, introduction to the formalities of mathematical logic and to the rules of inference that we will use in constructing our proofs. 1.1 Basic Notions of Mathematical Logic and Truth Tables A statement is a (mathematical) sentence that can be determined to be either true or false. For example, all differentible functions are continuous, and all prime numbers are odd are two examples of mathematical statements. The first statement will be later shown to be true and, since 2 is a natural number 1 that is both prime and even, the latter statement is false. Sometimes we are not able to determine whether or not a statement is true or false but we can see that it must be one or the other. For example, the Twin Prime Conjecture says that there are infinitely primes p such that p + 2 is also prime. Despite a great deal of effort that has been exerted to try and prove this statement, we still do not know that it is true. (A conjecture is a statement for which there is evidence or strong speculation that it is true but no known proof). However, it should be obvious that this statement is either true or it is false. A mathematical sentence such as x > 0 is not a statement since it can be either true or false depending on the value assigned to the variable x . Throughout the rest of this chapter we will use italicized lower case letters to denote statements. Given a statement p , we can also talk about the negation of p which we denote by p and which we call not p . The negation of a statement is exactly what one would expect from the name. For example, if the statement p is the sky is blue, then the negation p is simply the statement that the sky is not blue. When a statement p is true, its negation...
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