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Discrete Transition to Advanced Mathematics

Discrete Transition to Advanced Mathematics - 04 edition

ISBN13: 978-0534405182

Cover of Discrete Transition to Advanced Mathematics 04 (ISBN 978-0534405182)
ISBN13: 978-0534405182
ISBN10: 0534405185
Cover type:
Edition/Copyright: 04
Publisher: Brooks/Cole Publishing Co.
Published: 2004
International: No

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Discrete Transition to Advanced Mathematics - 04 edition

ISBN13: 978-0534405182

Bettina Richmond and Thomas Richmond

ISBN13: 978-0534405182
ISBN10: 0534405185
Cover type:
Edition/Copyright: 04
Publisher: Brooks/Cole Publishing Co.

Published: 2004
International: No
Summary

As the title indicates, this text is intended for courses aimed at bridging the gap between lower level mathematics and advanced mathematics. The transition to advanced mathematics presented is discrete since continuous functions are not studied. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples to develop an understanding of discrete mathematics and critical thinking skills. Including more topics than can be covered in one semester, the text offers innovative material throughout, particularly in the last three chapters (e.g. Fibonacci Numbers and Pascal's Triangle). This allows flexibility for the instructor and the ability to teach a deeper, richer course.

Benefits:

  • Provides, in a clear writing style, an intuitive understanding of concepts behind precise statements.
  • Demonstrates the remarkable connections between the Fibonacci numbers, Pascal's Triangle, and the golden ratio. These ideas serve as an excellent capstone to the course.
  • Includes interesting material for reading projects, such as Divisibility Tests (Section 3.4), Number Patterns (Section 3.5), Infinite Arithmetic (Section 8.5), and Fibonacci Numbers and Pascal's Triangle (Chapter 9).
  • Provides over 650 exercises of varying difficulty designed to reinforce and extend the material presented.
  • Contains many classic results, as well as many elegant or surprising results that are not widely known.

Author Bio

Richmond, Bettina : Western Kentucky University


Richmond, Thomas : Western Kentucky University

Table of Contents

1.SETS AND LOGIC

Sets
Set Operations
Partitions
Logic and Truth Tables
Quantifiers
Implications

2. PROOFS

Proof Techniques
Mathematical Induction
The Pigeonhole Principle

3. NUMBER THEORY

Divisibility
The Euclidean Algorithm
The Fundamental Theorem of Arithmetic
Divisibility Tests
Number Patterns

4. COMBINATORICS

Getting from Point A to Point B
The Fundamental Principle of Counting
A Formula for the Binomial Coefficients
Combinatorics with Indistinguishable Objects
Probability

5. RELATIONS

Relations
Equivalence Relations
Partial Orders
Quotient Spaces

6. FUNCTIONS AND CARDINALITY

Functions
Inverse Relations and Inverse Functions
Cardinality of Infinite Sets
An Order Relation for Cardinal Numbers

7. GRAPH THEORY

Graphs
Matrices, Digraphs, and Relations
Shortest Paths in Weighted Graphs
Trees

8. SEQUENCES

Sequences
Finite Differences
Limits of Sequences of Real Numbers
Some Convergence Properties
Infinite Arithmetic
Recurrence Relations

9. FIBONACCI NUMBERS AND PASCAL'S TRIANGLE

Pascal's Triangle
The Fibonacci Numbers
The Golden Ratio
Fibonacci Numbers and the Golden Ratio
Pascal's Triangle and the Fibonacci Numbers

10. CONTINUED FRACTIONS

Finite Continued Fractions
Convergents of a Continued Fraction
Infinite Continued Fractions
Applications of Continued Fractions

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