Mathematical Reasoning: Writing and Proof

by Sundstrom, Ted A.

• ISBN13: 9780130618153
• ISBN10: 0130618152
• Format: Hardcover
• Copyright: 2003-01-01
• Publisher: Pearson College Div

Summary

Focusing on the formal development of mathematics, this book demonstrates how to read and understand, write and construct mathematical proofs. It emphasizes active learning, and uses elementary number theory and congruence arithmetic throughout. Chapter content covers an introduction to writing in mathematics, logical reasoning, constructing proofs, set theory, mathematical induction, functions, equivalence relations, topics in number theory, and topics in set theory. For learners making the transition form calculus to more advanced mathematics.

Table of Contents

1. Introduction to Writing in Mathematics.
2. Logical Reasoning.
3. Constructing Proofs.
4. Set Theory.
5. Mathematical Induction.
6. Functions.
7. Equivalence Relations.
8. Topics in Number Theory.
9. Topics in Set Theory.
A: Guidelines for Writing Mathematical Proofs.
B: Answers and Hints for Selected Exercises.

Excerpts

Mathematical Reasoning: Writing and Proofis designed to be a text for to first course in the college mathematics curriculum that focuses on the development of mathematics. The primary goals of the text are as To help students learn how to read and understand mathematical definitions and proofs; To help students learn how to construct mathematical proofs; To help students learn how to write mathematical proofs according to accepted guidelines so that their work and reasoning can be understood by others; and To provide students with some mathematical material that will be needed for their further study of mathematics. This type of course is becoming a standard part of the mathematics major at most colleges and universities. It is often referred to as a "transition course" from the calculus sequence to the upper level courses in the major. The transition is from the problem-solving orientation of calculus to the more abstract and theoretical upper-level courses. This is needed today because the principal goals of most calculus courses are developing students' understanding of the concepts of calculus and improving their problem-solving skills. Consequently, most students complete their study of calculus without seeing a formal proof or having constructed a proof of their own. This is in contrast to many upper-level mathematics courses, where the emphasis is on the formal development of abstract mathematical ideas, and the expectations are that students will be able to read and understand proofs and to construct and write coherent, understandable mathematical proofs. Important Features of the Book Mathematical Reasoningwas written to assist students with the transition from calculus to upper level mathematics courses. Students should be able to use this text with a background of one semester of calculus. Following are some of the important ways this text will help with this transition. 1. Emphasis on Writing in Mathematics The issue of writing mathematical exposition is addressed throughout the book. Guidelines for writing mathematical proofs are incorporated into the text. These guidelines are introduced as needed and begin in Chapter 1. Appendix A contains a summary of all the guidelines for writing mathematical proofs that are introduced in the text. In addition, every attempt has been made to ensure that each proof presented in this text is written according to these guidelines in order to provide students with examples of well-written proofs. 2. Instruction in the Process of Constructing Proofs One of the primary goals of this book is to develop students' abilities to construct mathematical proofs. Another goal is to develop their abilities to write the proof in a coherent manner that conveys an understanding of the proof to the reader. These are two distinct skills. Instruction on how to write proofs begins in Section 1.2 and is developed further in Chapter 3. In addition, Chapter 5 is devoted to developing students' abilities to construct proofs using mathematical induction. Students are taught to organize their thought processes when attempting to construct a proof with a so-called know-show table. (See Sections 1.2 and 3.1.) Students use this table to work backward from what it is they are trying to prove while at the same time working forward from the assumptions of the problem. 3. Emphasis on Active Learning One of the underlying premises of this text is that the best way to learn and understand mathematics is to be actively involved in the learning process. However, it is unreasonable to expect students to go out and learn mathematics on their own. Students actively involved in learning mathematics need appropriate materials that will provide guidance and support in their learning of mathematics. This text provides these by incorporating two or three Preview Activities for each section and