Introduction to Mathematical Proofs: 2nd Edition (Hardback) book cover

Introduction to Mathematical Proofs

2nd Edition

By Charles Roberts

Chapman and Hall/CRC

414 pages | 51 B/W Illus.

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Hardback: 9781482246872
pub: 2014-12-17
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Description

Introduction to Mathematical Proofs helps students develop the necessary skills 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.

This new edition includes more than 125 new exercises in sections titled More Challenging Exercises. Also, numerous examples illustrate in detail how to write proofs and show how to solve problems. These examples can serve as models for students to emulate when solving exercises.

Several biographical sketches and historical comments have been included to enrich and enliven the text. 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 prepares them to succeed in more advanced mathematics courses, such as abstract algebra and analysis.

Table of Contents

Logic

Statements, Negation, and Compound Statements

Truth Tables and Logical Equivalences

Conditional and Biconditional Statements

Logical Arguments

Open Statements and Quantifiers

Chapter Review

Deductive Mathematical Systems and Proofs

Deductive Mathematical Systems

Mathematical Proofs

Chapter Review

Set Theory

Sets and Subsets

Set Operations

Additional Set Operations

Generalized Set Union and Intersection

Chapter Review

Relations

Relations

The Order Relations <, , >,

Reflexive, Symmetric, Transitive, and Equivalence Relations

Equivalence Relations, Equivalence Classes, and Partitions

Chapter Review

Functions

Functions

Onto Functions, One-to-One Functions and One-to-One Correspondences

Inverse of a Function

Images and Inverse Images of Sets

Chapter Review

Mathematical Induction

Mathematical Induction

The Well-Ordering Principle and the Fundamental Theorem of Arithmetic

Cardinalities of Sets

Finite Sets

Denumerable and Countable Sets

Uncountable Sets

Proofs from Real Analysis

Sequences

Limit Theorems for Sequences

Monotone Sequences and Subsequences

Cauchy Sequences

Proofs from Group Theory

Binary Operations and Algebraic Structures

Groups

Subgroups and Cyclic Groups

Appendix Reading and Writing Mathematical Proofs

Answers to Selected Exercises

References

Index

About the Author

Charles Roberts, PhD, professor, Department of Math and Computer Science, Indiana State University, Terre Haute, USA

About the Series

Textbooks in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT028000
MATHEMATICS / Set Theory
MAT037000
MATHEMATICS / Functional Analysis