A Bridge to Higher Mathematics 1st Edition by Valentin Deaconu, Donald Pfaff 9781498775250 149877525X

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Product details:

ISBN 10: 149877525X

ISBN 13: 9781498775250

Author: Valentin Deaconu, Donald C. Pfaff

A Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next is the construction of integers including some elementary number theory. The notions of finite and infinite sets, cardinality of counting techniques and combinatorics illustrate more techniques of proof. For more advanced readers, the text concludes with sets of rational numbers, the set of reals and the set of complex numbers. Topics, like Zorn’s lemma and the axiom of choice are included. More challenging problems are marked with a star. All these materials are optional, depending on the instructor and the goals of the course.

Table of contents:

Chapter 1: Elements of Logic

Chapter 2: Proofs: Structures and Strategies

Chapter 3: Elementary Theory of Sets

Chapter 4: Functions

Chapter 5: Relations

Chapter 6: Axiomatic Theory of Positive Integers

Chapter 7: Elementary Number Theory

Chapter 8: Cardinality: Finite Sets, Infinite Sets

Chapter 9: Counting Techniques and Combinatorics

Chapter 10: The Construction of Integers and Rationals

Chapter 11: The Construction of Real and Complex Numbers

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Tags: Valentin Deaconu, Donald Pfaff, Bridge, Mathematics