Introduction to Mathematical Proofs: A Transition

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CRC Press, Jun 24, 2009 - Mathematics - 433 pages
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Shows How to Read & Write Mathematical Proofs
Ideal Foundation for More Advanced Mathematics Courses

Introduction to Mathematical Proofs: A Transition facilitates a smooth transition from courses designed to develop computational skills and problem solving abilities to courses that emphasize theorem proving. It helps students develop the skills necessary to write clear, correct, and concise proofs.

Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It also covers elementary topics in set theory, explores various properties of relations and functions, and proves several theorems using induction. The final chapters introduce the concept of cardinalities of sets and the concepts and proofs of real analysis and group theory. In the appendix, the author includes some basic guidelines to follow when writing proofs.

Written in a conversational style, yet maintaining the proper level of mathematical rigor, this accessible book teaches students to reason logically, read proofs critically, and write valid mathematical proofs. It will prepare them to succeed in more advanced mathematics courses, such as abstract algebra and geometry.

 

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Contents

Logic
1
Deductive Mathematical Systems and Proofs
69
Set Theory
125
Relations
175
Functions
219
Mathematical Induction
261
Cardinalities of Sets
275
Proofs from Real Analysis
297
Proofs from Group Theory
325
Reading and Writing Mathematical Proofs
349
Answers to Selected Exercises
357
References
415
Index
417
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About the author (2009)

Charles E. Roberts, Jr. is a professor of mathematics and computer science at Indiana State University.

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