The Elements of Advanced Mathematics

Abstract

Basic Logic Principles of Logic Truth And and Or Not If-Then Contrapositive, Converse, and Iff Quantifiers Truth and Provability 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 Relations and Functions Relations Order Relations Functions Combining Functions Cantor's Notion of Cardinality Axioms of Set Theory, Paradoxes, and Rigor Axioms of Set Theory The Axiom of Choice Independence and Consistency Set Theory and Arithmetic Number Systems The Natural Number System The Integers The Rational Numbers The Real Number System The Nonstandard Real Number System The Complex Numbers The Quaternions, the Cayley Numbers, and Beyond More on the Real Number System Introductory Remark Sequences Open Sets and Closed Sets Compact Sets The Cantor Set A Glimpse of Topology What Is Topology? First Definitions Mappings The Separation Axioms Compactness Theoretical Computer Science Introductory Remarks Primitive Recursive Functions General Recursive Functions Description of Boolean Algebra Axioms of Boolean Algebra Theorems in Boolean Algebra Illustration of the Use of Boolean Logic The Robbins Conjecture The P/NP Problem Introduction The Complexity of a Problem Comparing Polynomial and Exponential Complexity Polynomial Complexity Assertions That Can Be Verified in Polynomial Time Nondeterministic Turing Machines Foundations of NP-Completeness Polynomial Equivalence Definition of NP-Completeness Examples of Axiomatic Theories Group Theory Euclidean and Non-Euclidean Geometry Zero-Knowledge Proofs Basics and Background Preparation for RSA The RSA System Enunciated The RSA Encryption System Explicated Zero-Knowledge Proofs Solutions to Selected Exercises Bibliography Index Exercises appear at the end of each chapter.