Proof Patterns

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.

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 Includes supplementary material: sn.pub/extras

Autorentext
Mark Joshi, a leading mathematician in analysis and financial mathematics, published several successful textbooks and over sixty research papers in mathematics. He is known for his innovative viewpoints and intuitive style.

Inhalt
Induction and complete induction.- Double Counting.- The pigeonhole principle.- Divisions.- Contrapositive and contradiction.- Intersection-enclosure and Generation.- Difference of invariants.- Linear dependence, fields and transcendence.- Formal equivalence.- Equivalence extension.- Proof by classification.- Specific-generality.- Diagonal tricks and cardinality.- Connectedness and the Jordan curve theorem.- The Euler characteristic and the classification of regular polyhedra.- Discharging.- The matching problem.- Games.- Analytical patterns.- Counterexamples.