Bashirov Mathematical Analysis Fundamentals

1. Auflage 2014
ISBN: 978-0-12-801050-1
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 362 Seiten

The author's goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options. - Friendly and well-rounded presentation of pre-analysis topics such as sets, proof techniques and systems of numbers - Deeper discussion of the basic concept of convergence for the system of real numbers, pointing out its specific features, and for metric spaces - Presentation of Riemann integration and its place in the whole integration theory for single variable, including the Kurzweil-Henstock integration - Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus

Autoren/Hrsg.

Bashirov, Agamirza

Leseproben

Chapter 1 Sets and Proofs
Abstract
Chapter 1 gently introduces the concept of set, operations on sets, and other related definitions. The chapter also includes elements of mathematical logic and basic proof techniques. A special attention is given to the structure of proofs. Relations and functions are defined from general point of view. Two important relations, equivalence and order relations are discussed. The chapter ends with a list of axioms of set theory and some starting implications from them. The proofs in this chapter are given with wide discussions and explanations. Keywords
Set; proposition; mathematical logic; relation; function; axioms of set theory Mathematics deals with a great variety of mathematical concepts, each having a precise mathematical definition. Unlike the explanatory method, adopted in monolingual dictionaries, mathematical definitions are hierarchical: each of them uses only those concepts that are defined previously. Running back along this hierarchy of mathematical definitions, after all, one can get a basic concept for all the others—the concept of a set. Since there is nothing in mathematics foregoing the primitive concept of a set, one should carefully work with sets by handling them under a system of axioms that excludes an occurrence of paradoxical cases and at the same time preserves a wide range of manipulations. In this chapter sets and set-theoretic concepts are discussed. Although a discussion of the axioms of set theory lies out of the scope of analysis here, they are briefly mentioned at the end of this chapter. Another feature of this chapter is an emphasis on techniques of proof. An ability to read and write mathematical proofs is very important to study analysis. Respectively, every proof given in this chapter is accompanied with a detailed discussion. 1.1 Sets, Elements, and Subsets
A set is a primitive concept of mathematics. Intuitively, it is understood as a collection of objects that are called its elements or members. Sometimes we prefer to call a set as a class, system, or family. It is more desirable to symbolize sets by capital letters and their elements by lowercase letters. In this connection, the different groups of lowercase letters are used to denote different kinds of elements. In the rest, we will seek to use the letters ,b,c,… for elements that are fixed for the problem under consideration (parameters), ,y,z,… for unknowns and for variables, ,g,h,… for functions, ,m,k,… for integers, ,q,r,… for elements of metric spaces, ,d,s,… for small values, etc., though we do not make any strict convention about these usages. Both symbols ?A and ?a mean that is an element of the set or the set contains as its element. Similarly, both ?A and ?a mean that is not an element of the set or the set does not contain as its element. A set can be given by listing all its elements between braces. For example, the set a,b,c} consists of the three elements ,b, and . Often we use sets identified in the form a,b,c,…} if there is no ambiguity with the elements mentioned by the three dots. A set can also be given as a collection of all elements of a certain set having a certain property. For example, the set a?A:ahas the propertyP} consists of all elements of the set that have the property . The symbol a:ahas the propertyP} may be used for the preceding set if it is clear what is . Notice that this symbol without any existing set may cause a contradiction. For example, a:ais a set} does not exist as a set. Every set is completely determined by knowledge of all its elements and by nothing else. This simple remark has a few useful consequences. First, it defines a criterion for equality of sets: two sets and are equal if they consist of the same elements; this is indicated as =B. Otherwise, we write ?B. Second, it implies that any rearrangement as well as any repetition of the elements do not change the set. Consequently, when we identify a set by listing its elements, we usually list only its distinct elements disregarding their order. For example, the symbols a,b} and b,a} represent the same set consisting of two distinct elements and (symbolically, ?b). If =b ( and are equal elements), then we write a} or b} instead of a,b} and b,a}. Finally, it notifies that before forming a set as a collection of objects, at first these objects must be available. Consequently, it follows that there is no set containing itself as an element of itself. For example, the expression ={A,a,b,c,…} does not define as a set. A set containing only one element is called a singleton. The symbol a} expresses the singleton, containing as its element. emptysetØ is a set that does not contain any element. One must distinguish and Ø}, the first of them being the empty set and the second a singleton. If and are two sets so that every element of is an element of , then we say that is a subset of or is a supset of ; this is indicated as ?BorB?A. Clearly, for every set , it is true that ?A and ?A. We say that is a proper subset of or is a proper supset of if ?B and ?B; this is indicated as ?BorB?A. If is not a subset of , then we write ?BorB?A. One must correctly use the symbols and (respectively, and ) and distinguish an element from a subset. For this, the following guides are useful: ?set,subset?set. In fact, ?A implies a}?A and vice versa. The symbols ,?,?, and are graphically demonstrated by Venn3 diagrams in Figure 1.1(a)–(d).
Figure 1.1 Venn diagrams. 1.2 Operations on Sets
By use of operations on sets, we form new sets. Let and be two sets. The union of and is the set consisting of all elements of and ; it is denoted by ?B, that is, ?B={a:a?Aora?B}. The intersection of and is the set consisting of all common elements of and ; the symbol nB is used for this set, that is, nB={a:a?Aanda?B}. The difference of and is the set consisting of all elements of that are not in ;A?B denotes this set, that is, ?B={a?A:a?B}. If ?S, then ?A is called the complement of in ; we use also the symbol c for this set if there is no ambiguity about . These four operations on sets are graphically demonstrated by Venn diagrams in Figure 1.1(e)–(h). If is a family of sets, then the union of this family, together with its symbol, is defined by A?SA={a:a?Afor at least oneA?S}, and the intersection by A?SA={a:a?Afor everyA?S}. The Cartesian4 product or, simply, product of the sets...