A Bridge to Abstract Mathematics (Mathematical Association of America Textbooks)

A Bridge to Abstract Mathematics will prepare the mathematical novice to explore the universe of abstract mathematics. Mathematics is a science that concerns theorems that must be proved within the constraints of a logical system of axioms and definitions rather than theories that must be tested, revised, and retested. Readers will learn how to read mathematics beyond popular computational calculus courses. Moreover, readers will learn how to construct their own proofs. The book is intended as the primary text for an introductory course in proving theorems, as well as for self-study or as a reference. Throughout the text, some pieces (usually proofs) are left as exercises. Part V gives hints to help students find good approaches to the exercises. Part I introduces the language of mathematics and the methods of proof. The mathematical content of Parts II through IV were chosen so as not to seriously overlap the standard mathematics major. In Part II, students study sets, functions, equivalence and order relations, and cardinality. Part III concerns algebra. The goal is to prove that the real numbers form the unique, up to isomorphism, ordered field with the least upper bound. In the process, we construct the real numbers starting with the natural numbers. Students will be prepared for an abstract linear algebra or modern algebra course. Part IV studies analysis. Continuity and differentiation are considered in the context of time scales (nonempty, closed subsets of the real numbers). Students will be prepared for advanced calculus and general topology courses. There is a lot of room for instructors to skip and choose topics from among those that are presented. Some notes on notation To the students For the professors Part I. The Axiomatic Method: 1. Introduction 2. Statements in mathematics 3. Proofs in mathematics Part II. Set Theory: 4. Basic set operations 5. Functions 6. Relations on a set 7. Cardinality Part III. Number Systems: 8. Algebra of number systems 9. The natural numbers 10. The integers 11. The rational numbers 12. The real numbers 13. Cantor's reals 14. The complex numbers Part IV. Time Scales: 15. Time scales 16. The Delta Derivative Part V. Hints: 17. Hints for (and comments on) the exercises Index. Of the Properties of the Nonnegative IntegersThe Integers -- Introduction: Integers as Equivalence Classes -- A Total Ordering of the Integers -- Addition of Integers -- Multiplication of Integers -- Embedding the Natural Numbers in the Integers -- Supplemental Exercises -- Summary of the Properties of the Integers -- The Rational Numbers -- Introduction: Rationals as Equivalence Classes -- A Total Ordering of the Rationals -- Addition of Rationals -- Multiplication of Rationals -- An Ordered Field Containing the Integers -- Supplemental Exercises Statements In Mathematics -- Proofs In Mathematics -- Basic Set Operations -- Functions -- Relations On A Set -- Cardinality -- Algebra Of Number Systems -- The Natural Numbers -- The Integers -- The Rational Numbers -- The Real Numbers -- Cantor's Reals -- The Complex Numbers -- Time Scales -- The Delta Derivative -- Hints For (and Comments On) The Exercises. Ralph Oberste-vorth, Aristides Mouzakitis, Bonita A. Lawrence. Includes Bibliographical References (p. 223) And Index. A Complete Guide To Mathematical Proof, With Plentiful Exercises And Examples. Ideal For Educators, And As Preparation For Students. Oberste-Vorth, Ralph W., Mouzakitis, Aristides, Lawrence, Bonita A.