## ABSTRACT

** Discrete Mathematics and Applications, Second Edition** is intended for a one-semester course in discrete mathematics. Such a course is typically taken by mathematics, mathematics education, and computer science majors, usually in their sophomore year. Calculus is not a prerequisite to use this book.

Part one focuses on how to write proofs, then moves on to topics in number theory, employing set theory in the process. Part two focuses on computations, combinatorics, graph theory, trees, and algorithms.

- Emphasizes proofs, which will appeal to a subset of this course market
- Links examples to exercise sets
- Offers edition that has been heavily reviewed and developed
- Focuses on graph theory
- Covers trees and algorithms

**I Proofs **

**Logic and Sets **

Statement Forms and Logical Equivalences

Set Notation

Quantifiers

Set Operations and Identities

Valid Arguments

**Basic Proof Writing **

Direct Demonstration

General Demonstration (Part 1)

General Demonstration (Part 2)

Indirect Arguments

Splitting into Cases

**Elementary Number Theory **

Divisors

Well-Ordering, Division, and Codes

Euclid's Algorithm and Lemma

Rational and Irrational Numbers

Modular Arithmetic and Encryption

**Indexed by Integers **

Sequences, Indexing, and Recursion

Sigma Notation

Mathematical Induction, An Introduction

Induction and Summations

Strong Induction

The Binomial Theorem

**Relations **

General Relations

Special Relations on Sets

Basics of Functions

Special Functions

General Set Constructions

Cardinality

**II Combinatorics**

**Basic Counting **

The Multiplication Principle

Permutations and Combinations

Addition and Subtraction

Probability

Applications of Combinations

Correcting for Overcounting

**More Counting **

Inclusion-Exclusion

Multinomial Coe□cients

Generating Functions

Counting Orbits

Combinatorial Arguments

**Basic Graph Theory **

Motivation and Introduction

Special Graphs

Matrices

Isomorphisms

Invariants

Directed Graphs and Markov Chains

**Graph Properties **

Connectivity

Euler Circuits

Hamiltonian Cycles

Planar Graphs

Chromatic Number

**Trees and Algorithms **

Trees

Search Trees

Weighted Trees

Analysis of Algorithms (Part 1)

Analysis of Algorithms (Part 2)

**A Assumed Properties of Z and R **

**B Pseudocode **

**C Answers to Selected Exercises **