Exploring Mathematics An Engaging Introduction to Proof 1st edition by John Meier, Derek Smith ISBN 1107128986 ‎ 978-1107128989

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ISBN 10: 1107128986
ISBN 13: ‎ 978-1107128989
Author: John Meier, Derek Smith

Exploring Mathematics gives students experience with doing mathematics – interrogating mathematical claims, exploring definitions, forming conjectures, attempting proofs, and presenting results – and engages them with examples, exercises, and projects that pique their interest. Written with a minimal number of pre-requisites, this text can be used by college students in their first and second years of study, and by independent readers who want an accessible introduction to theoretical mathematics. Core topics include proof techniques, sets, functions, relations, and cardinality, with selected additional topics that provide many possibilities for further exploration. With a problem-based approach to investigating the material, students develop interesting examples and theorems through numerous exercises and projects. In-text exercises, with complete solutions or robust hints included in an appendix, help students explore and master the topics being presented. The end-of-chapter exercises and projects provide students with opportunities to confirm their understanding of core material, learn new concepts, and develop mathematical creativity.

Exploring Mathematics An Engaging Introduction to Proof 1st Table of contents:

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 ofArithmetic

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

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