Symbolic Logic and the Real Number System: An Introduction to the Foundations of Number Systems

Contents Preface Chapter 1 Symbolic Logic 1.1 The Logical Connectives 1.2 Truth-Table Analysis 1.3 Propositions Constructed from Independent Propositions 1.4 The Disjunctive Normal Form 1.5 Tautologies and Valid Arguments 1.6 The Algebra of Propositions 1.7 Applications to Switching Networks 1.8 The Universal Quantifier 1.9 The Existential Quantifier 1.10 Propositions Involving Several Quantifiers Chapter 2 Set Theory 2.1 Sets 2.2 Russell's Paradox 2.3 Operations on Sets 2.4 The Algebra of Sets 2.5 Ordered n-Tuples 2.6 Mappings 2.7 Operations 2.8 Relations 2.9 Equivalence Relations and Partitions 2.10 Measure Chapter 3 The Axiomatic Method and Abstract Algebra 3.1 Algebraic Systems 3.2 The Axiomatic Method 3.3 Semigroups 3.4 Groups 3.5 More Examples of Groups 3.6 Rings; Integral Domains 3.7 Fields; Ordered Fields 3.8 Vector Spaces 3.9 Boolean Algebra Chapter 4 Number Systems 4.1 The Peano System 4.2 The Natural Number System 4.3 The System of Integers 4.4 The Rational Number System Chapter 5 The Real Number System 5.1 Sequences of Decimal Rationals 5.2 The Real Number System 5.3 Almost-Constant Sequences 5.4 The DR Operation 5.5 Properties of the DR Operation 5.6 Properties of the Real Number System 5.7 Roots and Exponents Chapter 6 The Limit Concept 6.1 The Bk and Rk Operations 6.2 Almost-Constant Sequences 6.3 Properties of the Limit Concept 6.4 A Useful Technique Appendix 1 Cardinal Numbers Appendix 2 The Complex Number System References Index