The Elements of Advanced Mathematics, Second Edition: (Textbooks in Mathematics)

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.N ew 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

Review :
"The book helps students to cross the mathematical bridge to more advanced studies in topics such as topology, abstract algebra, and real analysis. It builds the background, tools, and skills students need to meet the challenges of mathematical rigor, axiomatics, and proofs." - in Zentralblatt MATH