Mathematical Thinking and Writing: A Transition to Higher Mathematics
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Table of contents
- Cover
- Contentsvii
- Why Read This Book?xiii
- Prefacexv
- Chapter 0. Notation and Assumptions1
- 0.1 Set Terminology and Notation1
- 0.2 Assumptions5
- Part I: Foundations of Logic and Proof Writing11
- Chapter 1. Logic13
- 1.1 Introduction to Logic13
- 1.2 If-Then Statements20
- 1.3 Universal and Existential Quantifiers27
- 1.4 Negations of Statements31
- Chapter 2. Beginner-Level Proofs38
- 2.1 Proofs Involving Sets38
- 2.2 Indexed Families of Sets47
- 2.3 Algebraic and Ordering Properties of R53
- 2.4 The Principle of Mathematical Induction61
- 2.5 Equivalence Relations: The Idea of Equality68
- 2.6 Equality, Addition, and Multiplication inQ76
- 2.7 The Division Algorithm and Divisibility79
- 2.8 Roots and irrational numbers85
- 2.9 Relations In General90
- Chapter 3. Functions97
- 3.1 Definitions and Terminology97
- 3.2 Composition and Inverse Functions106
- 3.3 Cardinality of Sets110
- 3.4 Counting Methods and the Binomial Theorem118
- Part II: Basic Priniciples of Analysis131
- Chapter 4. The Real Numbers133
- 4.1 The Least Upper Bound Axiom134
- 4.2 Sets in R140
- 4.3 Limit Points and Closure of Sets143
- 4.4 Compactness146
- 4.5 Sequences in R149
- 4.6 Convergence of Sequences153
- 4.7 The Nested Interval Property160
- 4.8 Cauchy Sequences165
- Chapter 5. Functions of a Real Variable170
- 5.1 Bounded and Monotone Functions170
- 5.2 Limits and Their Basic Properties173
- 5.3 More on Limits180
- 5.4 Limits Involving Infinity182
- 5.5 Continuity187
- 5.6 Implications of Continuity195
- 5.7 Uniform Continuity200
- Part III: Basic Principles of Alegbra205
- Chapter 6. Groups207
- 6.1 Introduction to Groups207
- 6.2 Generated and Cyclic Subgroups215
- 6.3 Integers Modulo n and Quotient Groups220
- 6.4 Permutation Groups and Normal Subgroups227
- 6.5 Group Morphisms236
- Chapter 7. Rings243
- 7.1 Rings and Subrings243
- 7.2 Ring Properties and Fields249
- 7.3 Ring Extensions256
- 7.4 Ideals260
- 7.5 Integral Domains267
- 7.6 UFDs and PIDs273
- 7.7 Euclidean Domains279
- 7.8 Ring Morphisms287
- 7.9 Quotient Rings291
- Index299
The ability to construct proofs is one of the most challenging aspects of the world of mathematics. It is, essentially, the defining moment for those testing the waters in a mathematical career. Instead of being submerged to the point of drowning, readers of Mathematical Thinking and Writing are given guidance and support while learning the language of proof construction and critical analysis.
Randall Maddox guides the reader with a warm, conversational style, through the task of gaining a thorough understanding of the proof process, and encourages inexperienced mathematicians to step up and learn how to think like a mathematician. A student's skills in critical analysis will develop and become more polished than previously conceived. Most significantly, Dr. Maddox has the unique approach of using analogy within his book to clarify abstract ideas and clearly demonstrate methods of mathematical precision.
Randall Maddox guides the reader with a warm, conversational style, through the task of gaining a thorough understanding of the proof process, and encourages inexperienced mathematicians to step up and learn how to think like a mathematician. A student's skills in critical analysis will develop and become more polished than previously conceived. Most significantly, Dr. Maddox has the unique approach of using analogy within his book to clarify abstract ideas and clearly demonstrate methods of mathematical precision.