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.
Inhalt
Foreword
1 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 Diagrams
2 Elementary Mathematical Logic
Inductive Reasoning
Deductive Reasoning
Mathematical Sentences
Replacement and Solution Sets
Compound Statements
Conjunctions
Disjunctions
Negation
Conditional Statements
Biconditional Sentences
3 Switching Circuits
Connectiops in Series
Connection in Parallel
The Distributive Law
Obtaining Functions Direct From Truth Tables
Negation
Gates
4 Multibase Arithmetic
Notation
Conversion From Base 10
Approximation
Addition
Subtraction
Multiplication
Division
Fractions
Puzzles
5 Finite Systems
Clock Arithmetic
Subtraction
Additive Inverses
Multiplicative Inverses
Modular Arithmetic
Fractions
A More Formal Discussion
Other Finite Systems
Groups
Isomorphisms
Subgroups
6 Relations
Relation Diagrams
Types of Relations
Cartesian Product
Inverse Relations
Graphs
Correspondences
Functions
Inverse Functions
Composition of Functions
7 Linear Programming
Equations for Straight Lines
Optimisation
Extreme Points
A Further Constraint
Non-Linear Programming
8 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 Products
9 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
Addition
10 Tessellatons
Quadrilaterals
Quadrisides
Pentagons
Hexagons
Off Cuts
11 Motion Geometry
Radial Enlargement
Reflection
Axes of Symmetry
Composite Reflections
Rotations
12 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
Solutions
Appendix: Some Abbreviations and Definitions
Index