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by Bettina Richmond and Thomas Richmond

Edition: 04Copyright: 2004

Publisher: Brooks/Cole Publishing Co.

Published: 2004

International: No

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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.

**Richmond, Bettina : Western Kentucky University **

Richmond, Thomas : Western Kentucky University

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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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

Publisher Info

Publisher: Brooks/Cole Publishing Co.

Published: 2004

International: No

Published: 2004

International: No

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.

**Richmond, Bettina : Western Kentucky University **

Richmond, Thomas : Western Kentucky University

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