Introduction to Mathematical Proofs A TRANSITION TO ADVANCED MATHEMATICS

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.

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 ReviewRelations 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 ReviewMathematical InductionMathematical 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 SequencesMonotone 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

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