Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition)
As we mentioned in the prefaces of the first two editions, because the teaching of calculus in many colleges and universities has become more problem-oriented with added emphasis on the use of calculators and computers, the theoretical gap between the material presented in calculus and the mathematical background expected (or at least hoped for) in more advanced courses such as abstract algebra and advanced calculus has widened.
In an attempt to narrow this gap and to better prepare students for the more abstract mathematics courses to follow, many colleges and universities have introduced courses that are now commonly called transition courses. In these courses, students are introduced to problems whose solution involves mathematical reasoning and a knowledge of proof techniques and writing clear proofs. Topics such as relations, functions and cardinalities of sets are encountered throughout theoretical mathematics courses. In addition, transition courses often include theoretical aspects of number theory, abstract algebra, and calculus. This textbook has been written for such a course.
The idea for this textbook originated in the early 1980s, long before transition courses became fashionable, during the supervising of undergraduate mathematics research projects. We came to realize that even advanced undergraduates lack a sound understanding of proof techniques and have difficulty writing correct and clear proofs.
At that time, a set of notes was developed for these students. This was followed by the introduction of a transition course, for which a more detailed set of notes was written. The first edition of this book emanated from these notes, which in turn has led to a second edition and now this third edition.
While understanding proofs and proof techniques and writing good proofs are major goals here, these are not things that can be accomplished to any great degree in a single course during a single semester. These must continue to be emphasized and practiced in succeeding mathematics courses.
Since this textbook originated from notes that were written exclusively for undergraduates to help them understand proof techniques and to write good proofs, this is the tone in which all editions of this book have been written: to be student-friendly. Numerous examples of proofs are presented in the text. Following common practice, we indicate the end of a proof with the square symbol . Often we precede a proof by a discussion, referred to as a proof strategy, where we think through what is needed to present a proof of the result in question. Other times, we find it useful to reflect on a proof we have just presented to point out certain key details.We refer to a discussion of this type as a proof analysis. Periodically, problems are presented and solved, and we may find it convenient to discuss some features of the solution, which we refer to simply as an analysis. For clarity, we indicate the end of a discussion of a proof strategy, proof analysis, analysis or solution of an example with the diamond symbol .
A major goal of this textbook is to help students learn to construct proofs of their own that are not only mathematically correct but clearly written. More advanced mathematics students should strive to present proofs that are convincing, readable, notationally consistent, and grammatically correct. A secondary goal is to have students gain sufficient knowledge of and confidence with proofs so that they will recognize, understand, and appreciate a proof that is properly written.
As with the first two editions, the third edition of this book is intended to assist the student in making the transition to courses that rely more on mathematical proof and reasoning. We envision students would take a course based on this book after they have had a year of calculus (and possibly another course, such as elementary linear algebra). It is likely that, prior to taking this course, a student’s training in mathematics consisted primarily of doing patterned problems; that is, students have been taught methods for solving problems, likely including some explanation as to why these methods worked. Students may very well have had exposure to some proofs in earlier courses but, more than likely, were unaware of the logic involved and of the method of proof being used. There may have even been times when the students were not certain what was being proved.
Outline of the Contents
Since writing good proofs requires a certain degree of competence in writing, we have devoted Chapter 0 to writing mathematics. The emphasis of this chapter is on effective and clear exposition, correct usage of symbols, writing and displaying mathematical expressions, and using key words and phrases. Although every instructor will emphasize writing in his or her own way, we feel that it is useful to read Chapter 0 periodically throughout the course. It will mean more as the student progresses through the course. Among the additions to and changes in the second edition that resulted in this third edition are the following.
• More than 250 exercises have been added, many of which require more thought to solve.
• New exercises have been added dealing with conjectures to give students practice with this important aspect of more advanced mathematics.
• Additional examples have been provided to assist in understanding and solving new exercises.
• In a number of instances, expanded discussions of a topic have been given to provide added clarity. In particular, the important topic of quantified statements is introduced in Section 2.10 and then reviewed in Section 7.2 to enhance one’s understanding of this.
• A discussion of cosets and Lagrange’s theorem has been added to Chapter 13 (Proofs in Group Theory).
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