The Elements of Advanced Mathematics, Second Edition

The gap between the rote, calculational learning mode of calculus and ordinary differential equations and the more theoretical learning mode of analysis and abstract algebra grows ever wider and more distinct, and students' need for a well-guided transition grows with it. For more than six years, the bestselling first edition of this classic text has helped them cross the mathematical bridge to more advanced studies in topics such as topology, abstract algebra, and real analysis. Carefully revised, expanded, and brought thoroughly up to date, the Elements of Advanced Mathematics, Second Edition now does the job even better, building the background, tools, and skills students need to meet the challenges of mathematical rigor, axiomatics, and proofs. New in the Second Edition: Expanded explanations of propositional, predicate, and first-order logic, especially valuable in theoretical computer science A chapter that explores the deeper properties of the real numbers, including topological issues and the Cantor set Fuller treatment of proof techniques with expanded discussions on induction, counting arguments, enumeration, and dissection Streamlined treatment of non-Euclidean geometry Discussions on partial orderings, total ordering, and well orderings that fit naturally into the context of relations More thorough treatment of the Axiom of Choice and its equivalents Additional material on Russell's paradox and related ideas Expanded treatment of group theory that helps students grasp the axiomatic method A wealth of added exercises

Table of Contents:
BASIC LOGIC Principles of Logic Truth "And" and "Or" "Not" "If-Then" Contrapositive, Converse, and "Iff" Quantifiers Truth and Provability Exercises METHODS OF PROOF What is a Proof? Direct Proof Proof by Contradiction Proof by Induction Other Methods of Proof SET THEORY Undefinable Terms Elements of Set Theory Venn Diagrams Further ideas in Elementary Set Theory Indexing and Extended Set Operations Exercises RELATIONS AND FUNCTIONS Relations Order Relations Functions Combining Functions Cantor's Notion of Cardinality Exercises AXIOMS OF SET THEORY, PARADOXES, AND RIGOR Axioms of Set Theory The Axiom of Choice Independence and Consistency Set Theory and Arithmetic Exercises NUMBER SYSTEMS Preliminary Remarks The Natural Number System The Integers The Rational Numbers The Real Number System The Non-Standard Real Number System The Complex Numbers The Quaternions, The Cayley Numbers, and Beyond MORE ON THE REAL NUMBER SYSTEM Introduction Sequences Open Sets and Closed Sets Compact Sets The Cantor Set Exercises EXAMPLES OF AXIOMATIC THEORIES Introductory Remarks Group Theory Euclidean and Non-Euclidean Geometry Exercises SOLUTIONS TO SELECTED EXERCISES BIBLIOGRAPHY INDEX