Fuchs S. Introduction to Proofs and Proof Strategies 2023

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Fuchs S. Introduction to Proofs and Proof Strategies 2023

Textbook in PDF format Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof. The material revolves around possible strategies to approaching a problem without classifying 'types of proofs' or providing proof templates. Instead, it helps students develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow. Beginning by discussing familiar and fundamental topics from a more theoretical perspective, the book moves on to inequalities, induction, relations, cardinality, and elementary number theory. The final supplementary chapters allow students to apply these strategies to the topics they will learn in future courses. With its focus on 'doing mathematics' through 200 worked examples, over 370 problems, illustrations, discussions, and minimal prerequisites, this course will be indispensable to first- and second-year students in mathematics, statistics, and computer science. Instructor resources include solutions to select problems. Key Features Helps students discover possible strategies to approaching a problem and develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow Focuses on 'doing mathematics', rather than on mathematical logic and proof-templates, through 200 worked examples, 100 clarifying illustrations, discussions, and over 370 problems ranging from concept checks to full proofs Allows students time to adapt to the increased challenge of writing mathematical proofs and thinking mathematically before being exposed to higher level mathematical formalism Aids in the transition from lower-level mathematics, focused on computations, to advanced material, focused on theory and proofs, by explaining familiar and fundamental topics from a theoretical perspective Shows how readers can apply strategies they learned to more advanced topics by including supplemental chapters in elementary combinatorics, limits and continuity, complex numbers, and linear algebra Core Material Numbers, Quadratics and Inequalities Sets, Functions and the Field Axioms Informal Logic and Proof Strategies Mathematical Induction Bijections and Cardinality Integers and Divisibility Relations Additional Topics Elementary Combinatorics Preview of Real Analysis – Limits and Continuity Complex Numbers Preview of Linear Algebra Index