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Discrete Mathematics with Applications,9780495391326

Discrete Mathematics with Applications

by
Edition: 4th
Format: Hardcover
Pub. Date: 8/4/2010
Publisher(s): Cengage Learning

Summary

Susanna Epp's Discrete Mathematics with Applications, Fourth Edition provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision.

This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age.

Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper level mathematics courses.

Table of Contents

Speaking Mathematically
Variables
The Language of Sets
The Language of Relations and Functions
The Logic of Compound Statements
Logical Form and Logical Equivalence
Conditional Statements
Valid and Invalid Arguments
Application: Digital Logic Circuits
Application: Number Systems and Circuits for Addition
The Logic of Quantified Statements
Predicates and Quantified Statements I
Predicatesand Quantified Statements II
Statements with Multiple Quantifiers
Arguments with Quantified Statements
Elementary Number Theory and Methods of Proof
Direct Proof and Counterexample I: Introduction
Direct Proof and Counterexample II: Rational Numbers.
Direct Proof and Counterexample III: Divisibility
Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem
Direct Proof and Counterexample V: Floor and Ceiling
Indirect Argument: Contradiction and Contraposition
Indirect Argument: Two Classical Theorems
Application: Algorithms
Sequences, Mathematical Induction, and Recursion
Sequences
Mathematical Induction I
MathematicalInduction II
Strong Mathematical Induction and the Well-Ordering Principle
Application: Correctness of Algorithms
Defining Sequences Recursively
Solving Recurrence Relations by Iteration
Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients
General Recursive Definitions and Structural Induction
Set Theory
Set Theory: Definitions and the Element Method of Proof
Set Identities
Disproofs and Algebraic Proofs
Boolean Algebras, Russell's Paradox, and the Halting Problem
Properties of Functions
Functions Defined on General Sets
One-to-one, Onto, Inverse Functions
Composition of Functions
Cardinality, Sizes of Infinity, and Applications to Computability
Properties of Relations
Relations on Sets (add material about relational databases)
Reflexivity, Symmetry, and Transitivity
Equivalence Relations
Modular Arithmetic with Applications to Cryptography
Partial Order Relations
Counting
Counting and Probability
The Multiplication Rule
Counting Elements of Disjoint Sets: The Addition Rule
The Pigeonhole Principle
Counting Subsets of a Set: Combinations. r-Combinations with Repetition Allowed
Pascal's Formula and the Binomial Theorem
Probability Axioms and Expected Value
Conditional Probability, Bayes' Formula, and Independent Events
Graphs and Trees
Graphs: An Introduction
Trails, Paths, and Circuits
Matrix Representations of Graphs
Isomorphisms of Graphs
Trees: Examples and Basic Properties
Rooted Trees
Spanning Trees and a Shortest Path Algorithm
Analyzing Algorithm Efficiency
Real-Valued Functions of a Real Variable and Their Graphs
O-, ?-, and ?-Notations
Application: Efficiency of Algorithms I. Exponential and Logarithmic Functions: Graphs and Orders
Application: Efficiency of Algorithms II
Regular Expressions and Finite State Automata
Formal Languages and Regular Expressions
Finite-State Automata
Simplifying Finite-State Automata
Table of Contents provided by Publisher. All Rights Reserved.

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