Proof Patterns

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 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 de Get Access No Access 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