Fundamental Concepts of Mathematics

Preface to the Second Edition

Preface

Chapter 1 Numbers for Counting

Definition of Counting. Addition Positional Notation; Commutative and Associative Properties; Recursive Definition

Mathematical Induction. Inequality. Subtraction. Multiplication

Shortcuts in Multiplication. The Distributive Property

Prime Numbers; the Infinity of Primes

Division; Quotient and Remainder

Exponentiation; Representation in a Scale. The Counterfeit Penny Problem. Tetration

The Arithmetic of Remainders. Rings and Fields. The Fundamental Theorem of Arithmetic

The equation ax-by-1; the Measuring Problem and the Explorer Problem

Groups. Isomorphism. Cyclic Groups. Normal Subgroups; the Normalizer, the Center, the Factor Group

Semi-groups. The Word problem for Semi-groups and for Groups

Congruences. Format's Theorem. Tests for Divisibility

Tests for Powers

Pascal's Triangle; Binomial Coefficients

Ordinal Numbers; Transfinite Ordinals; Transfinite Induction

Chapter 2 Numbers for Profit and Loss and Numbers for Sharing

Positive and Negative Integers. Addition, Subtraction and Multiplication of Integers. The Ring of Integers

Inequalities

Numbers for Sharing. Addition, Multiplication and Division of Fractions

Inequalities. Enumeration of Fractions

Farey Series. Index Laws

The Field of Rational Numbers. Negative Indices; Fractional Indices. The Square Root of 2. The Extension Field x+y/2

Polynomials. The Remainder Theorem. Remainder Fields

Enumeration of Polynomials

Examples I

Solutions to Examples I

Chapter 3 Numbers Unending

Decimal Fractions; Terminating and Recurring Decimals

Addition, Subtraction and Multiplication of Decimals

Irrational Decimals. Positive and Negative Decimals

Examples II

Solutions to Examples II

Convergence; some Important Limits. Generalized Binomial Theorem

Examples III

Solutions to Examples III

Sequence for e. The Exponential Series

Continuity

Intervals. Limit Point. Closed Sets and Open Sets. Closure. Interior Points. Denumerable Sets. Finite Sets. Infinite Sets. Sequence. Null Sequence

Continuity. Functions. Function of a Function. Inverse Functions

Examples IV

Solutions to Examples IV

Integration; Increasing Functions. Integral of a Sum

Differentiation. Derivative of an Integral; of a Sum, Product, Quotient and Composite Function. The Exponential and Logarithmic Functions. The Logarithmic Series

The Circular Functions; the Evaluation of π

Examples V

Solutions to Examples V

Pretender Numbers. Dyadic Numbers. Pretender Difference and Convergence; Pretender Limit

Chapter 4 Sets and Truth Function

Union and Intersection of Sets. Distributive Law. Complement of a Set. Inclusion; Partial Order. Boolean Arithmetic; Axiomatic Theory

Sentence Logic. Truth Tables. Representing Functions

Switching Circuits. Three Pole and Four Pole Switches

Axiomatic Theory; Consistency, Completeness and Independence of the Axioms. The Deduction Principle. Truth Tables as a Decision Method for Sentence Logic

Impossibility of a Decision Method for Arithmetic. Incompleteness of Arithmetic

Hilbert's Tenth Problem

Examples VI

Solutions to Examples VI

Chapter 5 Networks and Maps

Connectivity. Networks. Königsberg Bridges Problem

Necessary and Sufficient Conditions for a Traversable Network

Euler's Formula. Characteristic of a Surface. The Regular Solids

Map Coloring. Two-color, Three-color, Four- and Five-color Maps

Maps on Anchor Rings; on Möbius Bands

Metric Spaces; Neighborhood, Open Set. Limit Point, Closure. Continuous Mappings

Topological Space; Open Sets, Neighborhoods, Closure

Continuous Mappings; Necessary and Sufficient Conditions

Chapter 6 Axiomatic Theory of Sets

Index

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