The elements of advanced mathematics
By: Steven G. Krantz.
Edition: 2nd ed.
Series: Studies in advanced mathematics
Publisher: Chapman & Hall
Published: 2002
Place of publication: Boca Raton, Fla
Language: eng
Number of pages: 214
Subjects: Mathematics
Notes: Includes bibliographical references (p. 209-210) and index.
Table of contents: title: Machine generated contents note: 1 Basic Logic 1; title: 1.1 Principles of Logic1; title: 1.2 Truth2; title: 1.3 "And" and "Or" 3 3; title: 1.4 "Not"6; title: 1.5 "If- Then"7; title: 1.6 Contrapositive, Converse, and "Iff"10; title: 1.7 Quantifiers13; title: 1.8 Truth and Provability17; title: Exercises21; title: 2 Methods of Proof 27; title: 2.1 What is a Proof?27; title: 2.2 Direct Proof28; title: 2.3 Proof by Contradiction33; title: 2.4 Proof by Induction36; title: 2.5 Other Methods of Proof40; title: Exercises43; title: 3 Set Theory 47; title: 3.1 Undefinable Terms47; title: 3.2 Elements of Set Theory48; title: 3.3 Venn Diagrams52; title: 3.4 Further Ideas in Elementary Set Theory53; title: 3.5 Indexing and Extended Set Operations55; title: Exercises57; title: 4 Relations and Functions 61; title: 4.1 Relations61; title: 4.2 Order Relations 64; title: 4.3 Functions 66; title: 4.4 Combining Functions 69; title: 4.5 Cantor's Notion of Cardinality 73; title: Exercises 85; title: 5 Axioms of Set Theory, Paradoxes, and Rigor 91; title: 5.1 Axioms of Set Theory91; title: 5.2 The Axiom of Choice 94; title: 5.2.1 Well Ordering94; title: 5.2.2 The Continuum Hypothesis95; title: 5.2.3 Zorn's Lemma95; title: 5.2.4 The Hausdorff Maximality Principle96; title: 5.2.5 The Banach-Tarski Paradox97; title: 5.3 Independence and Consistency97; title: 5.4 Set Theory and Arithmetic 99; title: Exercises101; title: 6 Number Systems 105; title: 6.0 Preliminary Remarks105; title: 6.1 The Natural Number System106; title: 6.2 The Integers111; title: 6.3 The Rational Numbers117; title: 6.4 The Real Number System124; title: 6.5 The Non-Standard Real Number System133; title: 6.5.1 The Need for Non-Standard Numbers133; title: 6.5.2 Filters and Ultrafilters133; title: 6.5.3 A Useful Measure133; title: 6.5.4 An Equivalence Relation134; title: 6.5.5 An Extension of the Real Number System134; title: 6.6 The Complex Numbers135; title: 6.7 The Quaternions, the Cayley Numbers, and Beyond139; title: Exercises141; title: 7 More on the Real Number System 147; title: 7.0 Introduction 147; title: 7.1 Sequences147; title: 7.2 Open Sets and Closed Sets149; title: 7.3 Compact Sets151; title: 7.4 The Cantor Set152; title: 7.4.1 Construction of a Remarkable Compact Set153; title: Exercises156; title: 8 Examples of Axiomatic Theories 159; title: 8.0 Introductory Remarks159; title: 8.1 Group Theory159; title: 8.2 Euclidean and Non-Euclidean Geometry168; title: Exercises180.