2nd Edition

Introduction to Mathematical Proofs

By Charles Roberts Copyright 2015
    416 Pages 51 B/W Illustrations
    by Chapman & Hall

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    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.

    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

    The Order Relations <, , >,
    Reflexive, Symmetric, Transitive, and Equivalence Relations
    Equivalence Relations, Equivalence Classes, and Partitions
    Chapter Review

    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
    Limit Theorems for Sequences
    Monotone Sequences and Subsequences
    Cauchy Sequences

    Proofs from Group Theory
    Binary Operations and Algebraic Structures
    Subgroups and Cyclic Groups

    Appendix Reading and Writing Mathematical Proofs

    Answers to Selected Exercises




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