Proof Patterns

✍ Scribed by Mark Joshi

• Publisher: Springer
• Year: 2015
• Tongue: English
• Leaves: 189
• Category: Library

✦ Synopsis

Equips students to recognise proof patterns across fields in pure mathematics
Reinforces each technique with end of chapter problems
Supports further research with extensive additional reading suggestions

This innovative textbook introduces a new pattern-based approach to learning proof methods in the mathematical sciences. Readers will discover techniques that will enable them to learn new proofs across different areas of pure mathematics with ease. The patterns in proofs from diverse fields such as algebra, analysis, topology and number theory are explored. Specific topics examined include game theory, combinatorics, and Euclidean geometry, enabling a broad familiarity.

The author, an experienced lecturer and researcher renowned for his innovative view and intuitive style, illuminates a wide range of techniques and examples from duplicating the cube to triangulating polygons to the infinitude of primes to the fundamental theorem of algebra. Intended as a companion for undergraduate students, this text is an essential addition to every aspiring mathematician’s toolkit.

Content Level » Upper undergraduate

Keywords » Combinatorics - Euclidean Geometry - Game Theory - Mathematics Education - Patterns - Proof - Pure Mathematics

Related subjects » Analysis - Geometry & Topology - Mathematics Education - Number Theory and Discrete Mathematics

✦ Table of Contents

Front Matter
Pages i-xiii

Book Chapter
Pages 1-9
Induction and Complete Induction

Book Chapter
Pages 11-17
Double Counting

Book Chapter
Pages 19-23
The Pigeonhole Principle

Book Chapter
Pages 25-31
Divisions

Book Chapter
Pages 33-41
Contrapositive and Contradiction
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Book Chapter
Pages 43-51
Intersection-Enclosure and Generation

Book Chapter
Pages 53-64
Difference of Invariants

Book Chapter
Pages 65-71
Linear Dependence, Fields and Transcendence

Book Chapter
Pages 73-80
Formal Equivalence

Book Chapter
Pages 81-95
Equivalence Extension

Book Chapter
Pages 97-103
Proof by Classification

Book Chapter
Pages 105-108
Specific-generality

Book Chapter
Pages 109-118
Diagonal Tricks and Cardinality

Book Chapter
Pages 119-125
Connectedness and the Jordan Curve Theorem

Book Chapter
Pages 127-136
The Euler Characteristic and the Classification of Regular Polyhedra

Book Chapter
Pages 137-141
Discharging

Book Chapter
Pages 143-145
The Matching Problem

Book Chapter
Pages 147-150
Games

Book Chapter
Pages 151-172
Analytical Patterns

Book Chapter
Pages 173-180
Counterexamples

Back Matter
Pages 181-190