Studying Rudin's Principles of Mathematical Analysis Through ...

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$Math Department Presentation$

4 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS 1.1 Introduction 1.1.1 Example: √ 2 is not rational Show that the equation p 2 = 2 is not satisfied by any rational numbers. Hint Use contradiction. Start with the assumption that p can be expressed as a rational number. 1.1.2 Remark: Gaps in Q Let A be the set of all positive rationals p such that p 2 < 2 and let B consist of all positive rationals p such that p 2 > 2. Show that A contains no largest number and B contains no smallest. In other words, show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If r < s then r < (r + s)/2 < s. The real number system fills these gaps. 1.1.3 Definition: Set If A is any set (whose elements may be numbers or any other objects), we write x ∈ A to indicate that x is a member (or an element) of A. If x is not a member of A, we write: x /∈ A. Now define empty set, nonempty, proper subset. 1.1.4 Definition: Q The set of all rational numbers will be denoted by Q. 1.2 Ordered Sets 1.2.1 Definition: Order Let S be a set. Define order. 1.2.2 Definition: Ordered Set Define an ordered set. 1.2.3 Definition: Bound Define bounded above and an upper bound.
1.3. FIELDS 5 1.2.4 Definition: Supremum & Infimum Define the least upper bound (supremum) and the greatest upper bound (infimum) of a set. 1.2.5 Examples: Bounds, Sup, and Inf (a) Consider the sets A and B of Example 1.1.1 as subsets of the ordered set Q. Comment on A and B in terms of their bounds in Q. (b) If α = sup E exists. Is it necessarily α ∈ E? Give an example. (c) Let E consist of all numbers 1/n where n=1, 2, 3, What are sup E and inf E? 1.2.6 Definition: The Least-upper-bound Property Define the least-upper-bound property. 1.2.7 Theorem: Sup & Inf Relation Suppose S is an ordered set with the least-uper-bound property, B ⊂ S, B is not empty, and B is bounded below. Let L be the set of all lower bounds of B. Prove that α = sup L exists in S, and α = inf B, and, in particular, inf B exists in S. 1.3 Fields 1.3.1 Definition: Field & Field Axioms What is a field? What are the field axioms? 1.3.2 Remarks (a) Is Q a field? (b) Do real and complex numbers form fields? 1.3.3 Proposition: Implications of the Addition Axioms Prove that the axioms for addition imply the following statements: (a) If x + y = x + z, then y = z. (b) If x + y = x, then y = 0. (c) If x + y = 0, then y = −x. (d) −(−x) = x.
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