Proof Patterns

✍ Scribed by Mark Joshi (auth.)

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 189
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xiii
Induction and Complete Induction....Pages 1-9
Double Counting....Pages 11-17
The Pigeonhole Principle....Pages 19-23
Divisions....Pages 25-31
Contrapositive and Contradiction....Pages 33-41
Intersection-Enclosure and Generation....Pages 43-51
Difference of Invariants....Pages 53-64
Linear Dependence, Fields and Transcendence....Pages 65-71
Formal Equivalence....Pages 73-80
Equivalence Extension....Pages 81-95
Proof by Classification....Pages 97-103
Specific-generality....Pages 105-108
Diagonal Tricks and Cardinality....Pages 109-118
Connectedness and the Jordan Curve Theorem....Pages 119-125
The Euler Characteristic and the Classification of Regular Polyhedra....Pages 127-136
Discharging....Pages 137-141
The Matching Problem....Pages 143-145
Games....Pages 147-150
Analytical Patterns....Pages 151-172
Counterexamples....Pages 173-180
Back Matter....Pages 181-190

✦ Subjects

Number Theory; Geometry; Combinatorics; Analysis; Topology; Mathematics Education