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### Summary

### Summary

h Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.

Here in this highly useful reference is the finest overview of finite and discrete math currently available, with hundreds of finite and discrete math problems that cover everything from graph theory and statistics to probability and Boolean algebra. Each problem is clearly solved with step-by-step detailed solutions.

DETAILS

- The PROBLEM SOLVERS are unique - the ultimate in study guides.

- They are ideal for helping students cope with the toughest subjects.

- They greatly simplify study and learning tasks.

- They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding.

- They cover material ranging from the elementary to the advanced in each subject.

- They work exceptionally well with any text in its field.

- PROBLEM SOLVERS are available in 41 subjects.

- Each PROBLEM SOLVER is prepared by supremely knowledgeable experts.

- Most are over 1000 pages.

- PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly.

TABLE OF CONTENTS

Introduction

Chapter 1: Logic

Statements, Negations, Conjunctions, and Disjunctions

Truth Table and Proposition Calculus

Conditional and Biconditional Statements

Mathematical Induction

Chapter 2: Set Theory

Sets and Subsets

Set Operations

Venn Diagram

Cartesian Product

Applications

Chapter 3: Relations

Relations and Graphs

Inverse Relations and Composition of Relations

Properties of Relations

Equivalence Relations

Chapter 4: Functions

Functions and Graphs

Surjective, Injective, and Bijective Functions

Chapter 5: Vectors and Matrices

Vectors

Matrix Arithmetic

The Inverse and Rank of a Matrix

Determinants

Matrices and Systems of Equations, Cramer''s Rule

Special Kinds of Matrices

Chapter 6: Graph Theory

Graphs and Directed Graphs

Matrices and Graphs

Isomorphic and Homeomorphic Graphs

Planar Graphs and Colorations

Trees

Shortest Path(s)

Maximum Flow

Chapter 7: Counting and Binomial Theorem

Factorial Notation

Counting Principles

Permutations

Combinations

The Binomial Theorem

Chapter 8: Probability

Probability

Conditional Probability and Bayes'' Theorem

Chapter 9: Statistics

Descriptive Statistics

Probability Distributions

The Binomial and Joint Distributions

Functions of Random Variables

Expected Value

Moment Generating Function

Special Discrete Distributions

Normal Distributions

Special Continuous Distributions

Sampling Theory

Confidence Intervals

Point Estimation

Hypothesis Testing

Regression and Correlation Analysis

Non-Parametric Methods

Chi-Square and Contingency Tables

Miscellaneous Applications

Chapter 10: Boolean Algebra

Boolean Algebra and Boolean Functions

Minimization

Switching Circuits

Chapter 11: Linear Programming and the Theory of Games

Systems of Linear Inequalities

Geometric Solutions and Dual of Linear Programming Problems

The Simplex Method

Linear Programming - Advanced Methods

Integer Programming

The Theory of Games

Index

WHAT THIS BOOK IS FOR

Students have generally found finite and discrete math difficult subjects to understand and learn. Despite the publication of hundreds of textbooks in this field, each one intended to provide an improvement over previous textbooks, students of finite and discrete math continue to remain perplexed as a result of numerous subject areas that must be remembered and correlated when solving problems. Various interpretations of finite and discrete math terms also contribute to the difficulties of mastering the subject.

In a study of finite and discrete math, REA found the following basic reasons underlying the inherent difficulties of finite and discrete math:

No systematic rules of analysis were ever developed to follow in a step-by-step manner to solve typically encountered problems. This results from numerous different conditions and principles involved in a problem that leads to many possible different solution methods. To prescribe a set of rules for each of the possible variations would involve an enormous number of additional steps, making this task more burdensome than solving the problem directly due to the expectation of much trial and error.

Current textbooks normally explain a given principle in a few pages written by a finite and discrete math professional who has insight into the subject matter not shared by others. These explanations are often written in an abstract manner that causes confusion as to the principle''s use and application. Explanations then are often not sufficiently detailed or extensive enough to make the reader aware of the wide range of applications and different aspects of the principle being studied. The numerous possible variations of principles and their applications are usually not discussed, and it is left to the reader to discover this while doing exercises. Accordingly, the average student is expected to rediscover that which has long been established and practiced, but not always published or adequately explained.

The examples typically following the explanation of a topic are too few in number and too simple to enable the student to obtain a thorough grasp of the involved principles. The explanations do not provide sufficient basis to solve problems that may be assigned for homework or given on examinations.

Poorly solved examples such as these can be presented in abbreviated form which leaves out much explanatory material between steps, and as a result requires the reader to figure out the missing information. This leaves the reader with an impression that the problems and even the subject are hard to learn - completely the opposite of what an example is supposed to do.

Poor examples are often worded in a confusing or obscure way. They might not state the nature of the problem or they present a solution, which appears to have no direct relation to the problem. These problems usually offer an overly general discussion - never revealing how or what is to be solved.

Many examples do not include accompanying diagrams or graphs, denying the reader the exposure necessary for drawing good diagrams and graphs. Such practice only strengthens understanding by simplifying and organizing finite and discrete math processes.

Students can learn the subject only by doing the exercises themselves and reviewing them in class, obtaining experience in applying the principles with their different ramifications.

In doing the exercises by themselves, students find that they are required to devote considerable more time to finite and discrete math than to other subjects, because they are uncertain with regard to the selection and application of the theorems and principles involved. It is also often necessary for students to discover those "tricks" not revealed in their texts (or review books) that make it possible to solve problems easily. Students must usually resort to methods of trial and error to discover these "tricks," therefore finding out that they may sometimes spend several hours to solve a single problem.

When reviewing the exercises in classrooms, instructors usually request students to take turns in writing solutions on the boards and explaining them to the class. Students often find it difficult to explain in a manner that holds the interest of the class, and enables the remaining students to follow the material written on the boards. The remaining students in the class are thus too occupied with copying the material off the boards to follow the professor''s explanations.

This book is intended to aid students in finite and discrete math overcome the difficulties described by supplying detailed illustrations of the solution methods that are usually not apparent to students. Solution methods are illustrated by problems that have been selected from those most often assigned for class work and given on examinations. The problems are arranged in order of complexity to enable students to learn and understand a particular topic by reviewing the problems in sequence. The problems are illustrated with detailed, step-by-step explanations, to save the students large amounts of time that is often needed to fill in the gaps that are usually found between steps of illustrations in textbooks or review/outline books.

The staff of REA considers finite and discrete math a subject that is best learned by allowing students to view the methods of analysis and solution techniques. This learning approach is similar to that practiced in various scientific laboratories, particularly in the medical fields.