Introduction to mathematical proofs : a transition

著者

書誌事項

Introduction to mathematical proofs : a transition

Charles E. Roberts, Jr.

（Textbooks in mathematics）

Chapman & Hall/CRC, c2010

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注記

Includes bibliographical references(p.415-416) and index

内容説明・目次

内容説明

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.

目次

Logic Statements, Negation, and Compound Statements Truth Tables and Logical Equivalences Conditional and Biconditional Statements Logical Arguments Open Statements and Quantifiers Deductive Mathematical Systems and Proofs Deductive Mathematical Systems Mathematical Proofs Set Theory Sets and Subsets Set Operations Additional Set Operations Generalized Set Union and Intersection Relations Relations The Order Relations <, =, >, = Reflexive, Symmetric, Transitive, and Equivalence Relations Equivalence Relations, Equivalence Classes, and Partitions Functions Functions Onto Functions, One-to-One Functions, and One-to-One Correspondences Inverse of a Function Images and Inverse Images of Sets 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

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