Mathematics

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Mathematical Proofs: A Transition to Advanced Mathematics
Gary Chartrand, Western Michigan University
Albert D. Polimeni, SUNY, College at Fredonia
Ping Zhang, Western Michigan University

ISBN-10: 0201710900
ISBN-13: 9780201710908

Publisher: Pearson
Copyright: 2003
Format: Cloth; 384 pp
Published: 05/28/2002

Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise providing solid introductions to relations, functions, and cardinalities of sets.

Proof Strategies. Proofs are often preceded by a discussion, encouraging students to think about the proof before they jump right into it.

Proof Analysis. Proofs are followed by discussions and reflected upon, to point out certain key details.

Practice in writing mathematics. Chapter 0 emphasizes effective and clear exposition, correct usage of symbols, writing and displaying mathematical expressions, and using key words and phrases.

Early introduction of Sets. Coverage of Sets in Chapter 1 helps prepare students for coverage of Logic in Chapter 2.

Early introduction of Logic. Much of the emphasis in Chapter 2 is on statements, implications, and an introduction to qualified statements. This helps present what is needed to get into proofs as quickly as possible.

Proof by Contradiction. An entire chapter is devoted to proof by contradiction.

Exercises. A wide variety of exercises are provided in the text. There are exercises that present students with statements, asking them to decide whether they are true or false. There are proposed proofs of statements, asking if the argument is valid. There are also proofs without a statement given, asking students to supply a statement of what has been proved.

Writing better proofs. A major goal of the text is to help students learn to construct proofs so that they are not only mathematically correct but are clearly written, convincing, readable, notationally consistent, and grammatically correct.

Web site for Mathematical Proofs. Three additional chapters, Chapters 14, 15, and 16 (dealing proofs in ring theory, linear algebra, and topology), can be found on the Web site: www.aw.com/chartrand.

0. Communicating Mathematics.

Learning Mathematics.

What Others Have Said About Writing.

Mathematical Writing.

Using Symbols.

Writing Mathematical Expressions.

Common Words and Phrases in Mathematics.

Some Closing Comments About Writing.

1. Sets.

Describing a Set.

Special Sets.

Subsets.

Set Operations.

Indexed Collections of Sets.

Partitions of Sets.

Cartesian Products of Sets.

2. Logic.

Statements.

The Negation of a Statement.

The Disjunction and Conjunction of Statements.

The Implication.