Exploring Mathematics by MEIER, JOHN -- Read -- Imperial Library of Trantor

Index

Cover Half-title page Cambridge Mathematical Textbooks Title Page Copyright page Dedication Contents Preface 1 Let’s Play!

1.1 A Direct Approach 1.2 Fibonacci Numbers and the Golden Ratio 1.3 Inductive Reasoning 1.4 Natural Numbers and Divisibility 1.5 The Primes 1.6 The Integers 1.7 The Rationals, the Reals, and the Square Root of 2 1.8 End-of-Chapter Exercises

2 Discovering and Presenting Mathematics

2.1 Truth, Tabulated 2.2 Valid Arguments and Direct Proofs 2.3 Proofs by Contradiction 2.4 Converse and Contrapositive 2.5 Quantifiers 2.6 Induction 2.7 Ubiquitous Terminology 2.8 The Process of Doing Mathematics 2.9 Writing Up Your Mathematics 2.10 End-of-Chapter Exercises

3 Sets

3.1 Set Builder Notation 3.2 Sizes and Subsets 3.3 Union, Intersection, Difference, and Complement 3.4 Many Laws and a Few Proofs 3.5 Indexing 3.6 Cartesian Product 3.7 Power 3.8 Counting Subsets 3.9 A Curious Set 3.10 End-of-Chapter Exercises

4 The Integers and the Fundamental Theorem of Arithmetic

4.1 The Well-Ordering Principle and Criminals 4.2 Integer Combinations and Relatively Prime Integers 4.3 The Fundamental Theorem of Arithmetic 4.4 LCM and GCD 4.5 Numbers and Closure 4.6 End-of-Chapter Exercises

5 Functions

5.1 What is a Function? 5.2 Domain, Codomain, and Range 5.3 Injective, Surjective, and Bijective 5.4 Composition 5.5 What is a Function? Redux! 5.6 Inverse Functions 5.7 Functions and Subsets 5.8 A Few Facts About Functions and Subsets 5.9 End-of-Chapter Exercises

6 Relations

6.1 Introduction to Relations 6.2 Partial Orders 6.3 Equivalence Relations 6.4 Modulo m 6.5 Modular Arithmetic 6.6 Invertible Elements 6.7 End-of-Chapter Exercises

7 Cardinality

7.1 The Hilbert Hotel, Count von Count, and Cookie Monster 7.2 Cardinality 7.3 Countability 7.4 Key Countability Lemmas 7.5 Not Every Set is Countable 7.6 Using the Schröder–Bernstein Theorem 7.7 End-of-Chapter Exercises

8 The Real Numbers

8.1 Completeness 8.2 The Archimedean Property 8.3 Sequences of Real Numbers 8.4 Geometric Series 8.5 The Monotone Convergence Theorem 8.6 Famous Irrationals 8.7 End-of-Chapter Exercises

9 Probability and Randomness

9.1 A Class of Lyin’ Weasels 9.2 Probability 9.3 Revisiting Combinations 9.4 Events and Random Variables 9.5 Expected Value 9.6 Flipped or Faked? 9.7 End-of-Chapter Exercises

10 Algebra and Symmetry

10.1 An Example from Modular Arithmetic 10.2 The Symmetries of a Square 10.3 Group Theory 10.4 Cayley Tables 10.5 Group Properties 10.6 Isomorphism 10.7 Isomorphism and Group Properties 10.8 Examples of Isomorphic and Non-isomorphic Groups 10.9 End-of-Chapter Exercises

11 Projects

11.1 The Pythagorean Theorem 11.2 Chomp and the Divisor Game 11.3 Arithmetic–Geometric Mean Inequality 11.4 Complex Numbers and the Gaussian Integers 11.5 Pigeons! 11.6 Mirsky’s Theorem 11.7 Euler’s Totient Function 11.8 Proving the Schröder–Bernstein Theorem 11.9 Cauchy Sequences and the Real Numbers 11.10 The Cantor Set 11.11 Five Groups of Order 8

Solutions, Answers, or Hints to In-Text Exercises Bibliography Index