Theorems, Corollaries, Lemmas, and Methods of Proof (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)

A hands-on introduction to the tools needed for rigorous and theoretical mathematical reasoning Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics. This essential book: Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs Reinforces the foundations of calculus and algebra Explores how to use both a direct and indirect proof to prove a theorem Presents the basic properties of real numbers/li> Discusses how to use mathematical induction to prove a theorem Identifies the different types of theorems Explains how to write a clear and understandable proof Covers the basic structure of modern mathematics and the key components of modern mathematics A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs. Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty. Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior- and senior-level courses on advanced calculus, real analysis, and modern algebra. Theorems, Corollaries, Lemmas, and Methods of Proof Contents Preface Chapter 1 — Introduction to Modern Mathematics 1.1 Inductive and Deductive Reasoning 1.2 Components of Modern Mathematics 1.3 Commonly Used Mathematical Notation EXERCISES Chapter 2 — An Introduction to Symbolic Logic 2.1 Statements and Propositional Functions 2.2 Combining Statements 2.3 Truth Tables 2.4 Conditional Statements 2.4.1 Converse and Contrapositive Statements 2.4.2 Biconditional Statements 2.5 Propositional Functions and Quantifiers EXERCISES Chapter 3 — Methods of Proof 3.1 Theorems, Corollaries, and Lemmas 3.2 The Contrapositive and Converse of a Theorem 3.3 Methods of Proof and Proving Theorems 3.3.1 Direct Proof 3.3.2 Indirect Proof 3.4 Specialized Methods of Proof 3.4.1 Mathematical Induction 3.4.2 Uniqueness Proofs 3.4.3 Existence Proofs 3.4.4 Proof by Cases 3.4.5 Proving Biconditional Theorems 3.4.6 Disproving a Conjecture 3.5 Some Final Notes on Proving Theorems EXERCISES Chapter 4 — Introduction to Number Theory 4.1 Binary Operators 4.2 Commonly Used Number Systems 4.2.1 The Natural Numbers 4.2.2 The Whole Numbers 4.2.3 The Integers 4.2.4 The Rational Numbers 4.2.5 The Real Numbers 4.3 Elementary Number Theory 4.3.1 Odd and Even Numbers 4.3.2 Divisibility 4.3.3 Prime Numbers 4.3.4 Recursively Defined Numbers EXERCISES Chapter 5 — The Foundations of Calculus 5.1 Functions 5.2 Sequences of Real Numbers 5.2.1 Convergent Sequences and Limit Theorems 5.2.2 Monotone Sequences 5.2.3 Cauchy Sequences 5.3 Limits of Functions 5.4 Continuity 5.5 Derivatives EXERCISES Chapter 6 — Foundations of Algebra 6.1 Introduction to Sets 6.1.1 Set Algebra 6.1.2 Element Chasing Proofs 6.1.3 Unions and Intersections of Finite Collections of Sets 6.1.4 Countable and Uncountable Sets 6.2 An Introduction to Group Theory 6.2.1 Groups 6.2.2 Subgroups EXERCISES References Index