Advanced Mathematics

A Transitional Reference

Stanley J. Farlow

• Publisher: Wiley
• Year: 2019
• ISBN:
  9781119563532
• Edition: 1
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

Provides a smooth and pleasant transition from first-year calculus to upper-level mathematics courses in real analysis, abstract algebra and number theory

Most universities require students majoring in mathematics to take a "transition to higher math" course that introduces mathematical proofs and more rigorous thinking. Such courses help students be prepared for higher-level mathematics course from their onset. Advanced Mathematics: A Transitional Reference provides a "crash course" in beginning pure mathematics, offering instruction on a blendof inductive and deductive reasoning. By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive setting.

Clear and concise chapters cover all the essential topics students need to transition from the "rote-orientated" courses of calculus to the more rigorous "proof-orientated" advanced mathematics courses. Topics include sentential and predicate calculus, mathematical induction, sets and counting, complex numbers, point-set topology, and symmetries, abstract groups, rings, and fields. Each section contains numerous problems for students of various interests and abilities. Ideally suited for a one-semester course, this book:

• Introduces students to mathematical proofs and rigorous thinking
• Provides thoroughly class-tested material from the authors own course in transitioning to higher math
• Strengthens the mathematical thought process of the reader
• Includes informative sidebars, historical notes, and plentiful graphics
• Offers a companion website to access a supplemental solutions manual for instructors

Advanced Mathematics: A Transitional Reference is a valuable guide for undergraduate students who have taken courses in calculus, differential equations, or linear algebra, but may not be prepared for the more advanced courses of real analysis, abstract algebra, and number theory that await them. This text is also useful for scientists, engineers, and others seeking to refresh their skills in advanced math.

Chapter 1
Logic and Proofs

1.1
Sentential Logic

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Purpose of Section

To introduce sentential (sen‐TEN‐shuhl) or propositional logic and the fundamental idea of a sentence (or proposition) and show how simple sentences can be combined using the logical connectives “and,” “or,” and “not” to form compound sentences. We then analyze the meaning of these connectives by means of truth tables and then introduce the concepts of logical equivalence, tautologies, and contradictions. We close by introducing conjunctive and disjunctive normal forms of logical expressions.

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1.1.1 Introduction

So what is mathematics? The word itself is derived from the Greek word mathēmatikē, meaning “knowledge” or “learning.” To both the practitioners of mathematics as well as the general public, the definition of the Queen of the Sciences varies widely.

• One of the greatest mathematician of Greek antiquity, Aristotle (384–322 BCE) defined mathematics as follows:

Mathematics is the science of quantity.

• Later, the Italian physicist Galileo (1564–1642), who was more interested in how it was applied, wrote:

“Mathematics is the language of the Universe and its characters are triangles, circles, and other geometric figures, without which it is impossible to understand a single word of it.”

• A more generic definition is g...