Modern Mathematics

Modern Mathematics: Made Simple presents topics in modern mathematics, from elementary mathematical logic and switching circuits to multibase arithmetic and finite systems. Sets and relations, vectors and matrices, tesselations, and linear programming are also discussed.Comprised of 12 chapters, this book begins with an introduction to sets and basic operations on sets, as well as solving problems with Venn diagrams. The discussion then turns to elementary mathematical logic, with emphasis on inductive and deductive reasoning; conjunctions and disjunctions; compound statements and conditional statements; and biconditional sentences. Subsequent chapters focus on switching circuits; multibase arithmetic; finite systems; relations, vectors, and matrices; tessellations; and linear programming. The book concludes with an analysis of motion geometry and rubber sheet geometry, paying particular attention to radial enlargement and composite reflections as well as topological equivalence, networks for maps, and incidence matrices.This monograph is intended for students, parents, and teachers who are interested in modern mathematics.

Foreword1 Sets Notation Numbers The Empty Set Finite and Infinite Sets Subsets Universal Sets Complement Intersection Union Basic Operations on Sets The Distributive Law Equations Solving Problems with Venn Diagrams2 Elementary Mathematical Logic Inductive Reasoning Deductive Reasoning Mathematical Sentences Replacement and Solution Sets Compound Statements Conjunctions Disjunctions Negation Conditional Statements Biconditional Sentences3 Switching Circuits Connectiops in Series Connection in Parallel The Distributive Law Obtaining Functions Direct From Truth Tables Negation Gates4 Multibase Arithmetic Notation Conversion From Base 10 Approximation Addition Subtraction Multiplication Division Fractions Puzzles5 Finite Systems Clock Arithmetic Subtraction Additive Inverses Multiplicative Inverses Modular Arithmetic Fractions A More Formal Discussion Other Finite Systems Groups Isomorphisms Subgroups6 Relations Relation Diagrams Types of Relations Cartesian Product Inverse Relations Graphs Correspondences Functions Inverse Functions Composition of Functions7 Linear Programming Equations for Straight Lines Optimisation Extreme Points A Further Constraint Non-Linear Programming8 Vectors The Triangle Law Vector Subtraction Multiplication of a Vector by a Scalar The Associative Law Vector Geometry Unit Vectors Perpendicular Unit Vectors The Dot Product The Distributive Law More Dot Products9 Matrices Reflections in the Axes Reflection in y = ±x Matrices for Reflections Dot Products Rotations Radial Expansion Matrix Multiplication The (2 ÷ 2) Inverse Matrix Simultaneous Equations A Group of Matrices Addition10 Tessellatons Quadrilaterals Quadrisides Pentagons Hexagons Off Cuts11 Motion Geometry Radial Enlargement Reflection Axes of Symmetry Composite Reflections Rotations12 Rubber Sheet Geometry Topological Equivalence Networks Euler's Results Networks for Maps Incidence Matrices V + F - E = 2 Surfaces Experiments in Topology: The Moebius Strip The Punctured Torus The Four-Colour Problem SolutionsAppendix: Some Abbreviations and DefinitionsIndex