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The Joy of Finite Mathematics
The Language and Art of Math

The Joy of Finite
Mathematics
The Language and Art of Math
Chris Tsokos
University of South Florida,
Department of Mathematics and Statistics,
Tampa, FL 33620
Rebecca Wooten
The Pedagogue, LLC,
Developing Educational Materials,
Tampa, FL
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier

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by the Publisher (other than as may be noted herein).
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Dedications
“We think faster than we speak and speak faster than we write; therefore,
when creating great things we abbreviate everything. This abbreviated
language is called Math.”
RDW
This text is dedicated to my wife Debbie and my children Matthew,
Jonathan and Maria.
Chris P. Tsokos
This text is dedicated to my husband Dana Miller, and all my family.
Rebecca D. Wooten

About the Authors
Chris P. Tsokos is Distinguished University Professor of Mathematics and Statistics at the University of South Florida.
Dr. Tsokos received his B.S. in Engineering Sciences/Mathematics, his M.A. in Mathematics from the University of Rhode
Island, and his Ph.D. in Statistics and Probability from the University of Connecticut. Professor Tsokos has also served on the
faculties at Virginia Polytechnic Institute and State University and the University of Rhode Island.
Dr. Tsokos’ research has extended into a variety of areas, including stochastic systems, statistical models, reliability
analysis, ecological systems, operations research, time series, Bayesian analysis, and mathematical and statistical modeling
of global warming, both parametric and nonparametric survival analysis, among others. He is the author of more than 300
research publications in these areas.
For the past four years Professor Tsokos’ research efforts have been focused on developing probabilistic models, para-
metric and nonparametric statistical models for cancer and GLOBAL WARMING data. Specifically, his research aims are
data driven and are oriented toward understanding the behavior of breast, lung, brain, and colon cancers. Information on the
subject matter can be found on his website.
Professor Tsokos has more than 300 publications in his research areas of interest. He is the author of several research
monographs and books, including Random Integral Equations with Applications to Life Sciences and Engineering, Prob-
ability Distribution: An Introduction to Probability Theory with Applications, Mainstreams of Finite Mathematics with
Applications, Probability with the Essential Analysis, Applied Probability Bayesian Statistical Methods with Applications
to Reliability, and Mathematical Statistics with Applications, among others.
Dr. Tsokos is the recipient of many distinguished awards and honors, including Fellow of the American Statistical Asso-
ciation, USF Distinguished Scholar Award, Sigma Xi Outstanding Research Award, USF Outstanding Undergraduate
Teaching Award, USF Professional Excellence Award, URI Alumni Excellence Award in Science and Technology, Pi
Mu Epsilon, election to the International Statistical Institute, Sigma Pi Sigma, USF Teaching Incentive Program, and several
humanitarian and philanthropic recognitions and awards.
Professor Tsokos is a member of several academic and professional societies. He is serving as Honorary Editor, Chief-
Editor, Editor or Associate Editor of more than twelve academic research journals.
Rebecca D. Wooten is Assistant Professor of Mathematics and Statistics at the University of South Florida. She received her
M.A./B.A. in Mathematics and her Ph.D. in Statistics from the University of South Florida. She has worked for 15 years in
teaching and has been recognized for her excellence in teaching; teaching courses such as Liberal Arts Math, Finite Math-
ematics, Basic Statistics, Introduction to Statistics, and Applied Statistics Methods.
Her research interests are concentrated in Applied Statistics with emphasis on Environmental Studies. Her research pub-
lications span a variety of areas such as Global Warming (carbon dioxide and temperature), Atmospheric Sciences and
Geography (hurricanes), Geology (volcanic ash fall), Marine Biology (red tide), among others.
Professor Wooten is extensively involved in activities to improve education not only in Mathematics and Statistics, but
Education in general. She is the Academic Coordinator for two free-educational assistance program which offer opportu-
nities for students to volunteer and the local community to get the assistance in their studies that they would otherwise be
unable to afford.
xi

Preface
This book has been written to present certain aspects of modern finite mathematics from an elementary point of view, with
emphasis on relevance to real-world problems. The objective is to create a positive attitude toward mathematics for the non-
science-orientated college student and to demonstrate its usefulness in solving problems that we frequently encounter in our
complex society.
Throughout the text, the aim has been to de-emphasize difficult theoretical concepts; thus, an intuitive treatment leads to
practical applications of the various subject area topics. We believe that with such an approach, the modern college student
will complete this course with the good feeling that mathematics is not only useful but enjoyable to work with.
The Joy of Finite Mathematics has several distinguishing features:
The text has been written for students with only high school mathematics.
Diagrams and graphs are used to illustrate mathematical concepts or thoughts.
Step-by-step directions are given for the implementation of mathematical methods to problem solving.
Emphasis has been placed on usefulness of mathematics to real-world problems.
To provide motivation to the reader, most chapters are preceded by a short biography of a scientist who made
important contributions to the subject area under consideration.
Mathematical concepts are introduced as clearly and as simply as possible, and they are followed by one or more
examples as an aid to thorough understanding.
Each chapter ends with a complete summary that includes the definitions, properties, and rules of the chapter, fol-
lowed by a Review Test.
Each chapter contains numerous critical thinking and basis exercises with problems that reflect on the mainstream of
the chapter.
The book has been designed to give the instructor wide flexibility in structuring a one or two-semester course, or a full-
year course. Although some chapters are dependent on other others, many options are allowed (see accompanying diagraph).
xiii

Chapter 1
Usefulness of Mathematics
Chapter 2
Logic
Chapter 3
Sets
Chapter 4
Counting
Chapter 5
Basic
Probabilty
Chapter 6
Binomial
Probabilty
Chapter 7
Normal
Probabilty
Chapter 8
Statistics
Chapter 9
Geometry
Chapter 10
Arithmetic &
Basic Algebra
Chapter 12
Game Theory
Chapter 11
Finance
xiv Preface

Very Special Acknowledgment
We wish to express our appreciation to the following academic educators for having reviewed our book and expressed their
opinions.
An outstanding book for students to obtain basic knowledge of the usefulness of mathematics. Excellent motivation strategies
throughout the book. It will inspire the student to learn the importance of mathematics.
Dr. Ram Kafle, Department of Mathematics and Statistics, Sam Houston University.
A very constructive and motivating book of finite mathematics. Special emphasis on the applications of math to real-world problems.
The interactive approach of presenting their material is excellent. The student will acquire a very good understanding of what math-
ematics is all about.
Dr. Bong-jin Choi, Lineberger Comprehensive Cancer Center, The University of North Carolina at Chapel Hill.
This book is a masterful treatment of finite mathematics for undergraduate students who are afraid of mathematics. It will enlighten
the student of the interdisciplinary use of mathematics at the very basic level. The book provides excellent illustrations of the use of
mathematics/statistics to solve important problems.
Dr. Yong Xu, Department of Mathematics and Statistics, Radford University.
The Joy of Finite Mathematics provides an excellent treatment of the subject. Unique emphasis on the importance of mathematical
sciences to our society. The non-mathematics-oriented undergraduate student will find the contents of the book easy to read and very
inspiring to learn more of the subject matter.
Dr. K. Pokhrel, Department of Mathematics & Computer Systems, Mercyhurst University.
This is an excellent book of finite mathematics. It offers a justifiable, useful, and motivating approach to what mathematics is all about
to the undergraduate student with minimum prior knowledge of the subject. The selection of the contents of the book, examples, and
exercises is outstanding.
Dr. N. Khanal, Department of Mathematics, University of Tampa.
xv

Several Options for a Semester Course
in Finite Mathematics
Five possible options in designing a basic course in finite mathematics are given below, along with some remarks for each
selection.
Options 1 and 2 offer a detailed coverage of specific topics in math, each spanning seven chapters:
Option 1:
Chapters Covered Title
Chapter 1 The Usefulness of Mathematics
Chapter 2 Logic
Chapter 3 Sets
Chapter 4 Counting Techniques
Chapter 5 Probability
Chapter 8 Statistics
Chapter 9 Geometry
Covering materials necessary for the CLAST (College Level Academic Skill Test) exam, excluding algebra, these six
topics are often taught collectively. In addition to the necessary high school algebra, these topics prepare a student well for
the CLAST exam.
Option 2:
Chapter 2 Logic
Chapter 3 Sets
Chapter 6 Bernoulli Trials
Chapter 7 The Bell-shaped Curve
Option 2 provides the materials necessary for a comprehensive understanding of basic probability and statistics. This
option is a broad introduction, including the underlying probabilities necessary to compute basic descriptive statistics, as
well as inferential statistics in terms of interval estimates and tests of hypothesis.
xvii

Options 3-5 offer a more detailed coverage of specific topics in math, each spanning six chapters:
Option 3:
Chapter 3 Sets
These topics enhance the study of probability. Option 3 begins with the basic concepts of categorization into sets,
counting sets, and measuring basic probabilities empirically. It then continues with measuring basic probabilities hypo-
thetically using either the discrete binomial probability distribution, or the continuous normal probability distribution.
Option 4:
Option 4 covers materials necessary for the study of the basic aspects of statistics. This option includes counting basic
empirical and hypothetical probabilities empirically. It also includes the basic necessities of statistics, descriptively and
inferentially, for means and proportions.
Option 5:
Chapter 2 Logic
Chapter 3 Sets
Chapter 9 Geometry
Chapter 11 Arithmetic and Algebra
Option 5 covers materials necessary to gain a basic understanding of the language of deterministic math. This option
provides a basic understanding of logic, sets, counting, geometry, and algebra.
Note: Game theory can be included in any scheme that includes the algebra and arithmetic.
xviii Several Options for a Semester Course in Finite Mathematics

A SUMMARY OF THE PROPOSED OPTIONS
Depending on which option you choose (1, 2, 3, 4, or 5), the purple indicates which chapters should be included; the green
indicates optional chapters in each scheme.
Several Options for a Semester Course in Finite Mathematics xix

Joy of Finite Mathematics
Special Features
Motivation
The usefulness of mathematics, especially those branches that constitute finite math, is illustrated both from a historical
perspective, and by the role it plays in our daily lives.
We emphasize an interactive approach to teaching finite mathematics.
The Language
Teaching any student basic finite mathematics requires a basic understanding of the underlying symbolic language.
Mathematics has many dialects: logic, set theory, combinatorics (counting), probability, statistics, geometry,
algebra, and finance, for example. Learning through relevance and interpretation of symbolism is vital.
The Relevant Questions
A complete introduction of mathematics in a finite world, the notation used, and the underlying interpretation is pre-
sented. Relevant and useful questions associated with each dialect are posed, which will be answered through the process
of learning finite mathematics.
The Review
Reviews of each basic concept are given at the end of each chapter. The reviews enhance the learning of the basic aspects
of each topic and their usefulness.
Step-by-Step
Clear and concise step-by-step procedures are used in the development of various methodologies. Procedures are easy to
follow, comprehend, and use to solve problems.
Highlights
Definitions, rules, methods and procedures are highlighted with boldface and their meanings and usefulness follow
with an abundance of relevant examples and applications.
Graphs and Tables
Throughout the book, emphasis is placed on the extensive use of tables, diagrams, and graphs to clearly illustrate
definitions, outlined methods, comparisons, etc. These visual aids invite clear interpretation of what they represent
and their relevance to the text.
Applications and Interpretation
We utilize a step-by-step approach in the illustrated examples (applications) that relate to the various dialects and their
interpretations that have been introduced. Emphasis is placed on properly denoting the problems symbolically, inter-
preting the argument, outlining the defined set, measuring the probability, or, in general, finding the solution. Then, we
encourage the student to clearly state any conclusions that can be drawn from the application.
xxi

Critical Reviews
Each chapter ends with a review of: the new mathematical vocabulary, the most important concepts and methods, an
abundance of review exercises, and a practice test that is based on the material from the preceding chapter.
Inspiration
Throughout the book, we utilize important historical facts and pose interesting and relevant questions. We also
include humorous events, pictures, graphs, tables, biographical sketches of famous scientists, popular and classical
quotes, and more. These are all tools to challenge, inspire and motivate students to learn the mathematical thinking and
to illustrate the absolute relevance of math to our society.
Challenging Problems
Throughout the book, there are sections and challenging problems that are somewhat more advanced for a basic course
in finite mathematics and are left to the discretion of the instructor.
xxii Joy of Finite Mathematics

1.1 Introduction to Math 3
1.2 What Is Logic? 5
1.3 Usefulness of Sets 6
1.4 Counting Techniques 6
1.5 Probability 7
1.6 Bernoulli Trials 8
1.7 The Bell-Shaped Curve 8
1.8 Statistics 9
1.9 Geometry 10
1.10 Arithmetic and Algebra 10
1.11 Finance 11
1.12 Game Theory 12
Critical Thinking and Basic Exercise 12
Summary of Important Concepts 13
Number is the within of all things.
PYTHAGORAS
Copyright © 2016 Elsevier Inc. All rights reserved. 1

Goals and Objectives
The main objective of this chapter is to give an overview and motivate the
non-mathematically oriented student about the usefulness of mathematics in
several important fields. We begin with a brief historical perspective of the
subject and proceed to discuss the importance and usefulness of all the areas
that we believe constitute a course in Finite Mathematics. The diagram
below illustrates the areas covered. Although not all the chapters of the
textbook need to be covered in a one semester or two quarter course, we
believe that the student can gain some basic knowledge by studying this
chapter.
Math
Logic
Sets
Counting
Probability
Bernuolli
Trials
Bell-shaped
Curve
Statistics
Geometry
Finance
Arithematic
& Algebra
Game
Theory
Thus, our goal here is to familiarize you with different areas and “dialects”
of Math and:
Learn about the history of Math
Learn about the Math that is the foundation of logic
Learn about the interplay of Math and sets
Learn about counting techniques
Learn about Math used to obtain probabilities of events
Learn about binomial trials that leads to the Bernoulli probabilities
Learn about the paramount importance of the Bell-Shaped Curve
Learn about using Math to develop useful statistical methods
If people do not believe that mathematics is simple,
it is only because they do not realize how
complicated life is.
John Louis von Neumann
The essence of mathematics is not to make simple
things complicated, but to make complicated things
simple.
S. Gudder
Go down deep enough into anything and you will
find mathematics.
Dean Schlicter
The man ignorant of mathematics will be
increasingly limited in his grasp of the main forces
of civilization.
John Kemeny
Pure mathematics is, in its way, the poetry of logical
ideas.
Albert Einstein
Mathematics is a more powerful instrument
of knowledge than any other that has
been bequeathed to us by human
agency.
Descartes
Mathematics is the Queen of the
Sciences.
Carl Friedrich Gauss
Mathematics is the science of definiteness,
the necessary vocabulary of those
who know.
W.J. White
Mathematics is the science which uses easy words
for hard ideas.
Edward Kasner and James R. Newman
Philosophy is a game with objectives and no rules.
Mathematics is a game with rules and no objectives.
2 Chapter 1 The Usefulness of Mathematics

Learn how Math is used to obtain measures of the earth or geometry
Learn the Math that is arithmetic and algebra
Learn how we use Math to answer basic financial questions
Learn how we use Math in Game Theory, solving systems of equations to
optimize strategies.
Pythagoras of Samos was the first to call himself a philosopher, Greek for “lover of wisdom.” Pythagorean ideas greatly
influenced western philosophy. Best known for the theorem which carries his name, Pythagoras was also a mathematician,
scientist, musician, and mystic.
He founded the religious movement called Pythagoreanism. The Pythagoreans first applied themselves to mathematics, a
science which they improved, and penetrated within; they fancied that the principles of mathematics were the principles of
all things. A younger contemporary, Eudemus, shrewdly remarked that “they changed geometry into a literal science; they
diverted arithmetics from the service of commerce”… Aristotle.
1.1 INTRODUCTION TO MATH
Mathematics played a very significant role in all our technological, scientific,
medical, educational and economic accomplishments in our global society.
However, just as important is the fact that mathematics indirectly interweaves
every aspect of our daily lives; mathematics is the most powerful interdisci-
plinary language in almost all fields of engineering, every aspect of health
sciences, education, social and physical sciences, economics, finance, envi-
ronmental sciences, Global Warming, and of course music and art, among
many other disciplines.
The word mathematics comes from the Greek word matheno which means
I learn. Historically mathematics has its origin in the Orient when the Baby-
lonians, in about 2000 BC, collected a lot of materials on the subject that we
identify as elementary algebra. However, the modern concept of mathematics
started in Greece around the fifth and fourth centuries BC. At this time, math-
ematics was subjected to philosophical discussion that was a unique priority in
the Greek city states. The Greek philosophers were quite aware of the math-
ematical difficulties involved in understanding continuity, infinity, motion and
the problems of making measurements of arbitrary quantities. Eudoxus’ theory
was very significant in geometrically understanding these concepts that were
later significantly improved by Euclid’s elements. Thus, the Greeks have an
enormous influence on the tremendous development of today’s mathematics.
Math is the language of thought. We think faster
than we speak and we speak faster than we write…
therefore, to convey our thoughts quickly,
Mathematicians abbreviate everything.
Rebecca D. Wooten
1.1 Introduction to Math 3

H. Weyl, one of the truly great mathematicians of the twentieth century, stated
“without the concepts, methods, and results found and developed by previous
generations right down to the Greek antiquity, one cannot understand either
the aim or the achievements of mathematics in the last 50 years” (American
Math Monthly, Vol. 102, 1995).
Historically mathematics was defined as the logical study of shape,
arrangement, and quantity. Furthermore, attempts have been made to think
of mathematics as two branches: Applied Mathematics and Pure or
Abstract Mathematics. The branches of applied mathematics are concerned
with the study of physical, biological, medical or sociological worlds. Pure or
Abstract Mathematics is concerned with the study and development of the
principles of mathematics as such and is not concerned with their immediate
usefulness.
In addition, we also had a divide of mathematics in the discrete and
the continuous. Herbert W. Turnbull, in his essay on the “World of Math-
ematics” states: To Pythagoras we owe the very word mathematics and its
double fold branches; that is,
Mathematics
The Discrete
The
absolute
Arithmetics
The
relative
Music
The Continued
The Stable
Geometry
The
Moving
Astronomy
This double fold of mathematics played a major role in the development
and usefulness of mathematics. In fact, Aristotle summarizes this historical
divide as follows:
“The Pythagoreans first applied themselves to mathematics, a science which
they improved; and, penetrated with it, they fancied that the principles of math-
ematics were the principles of all things.” And a younger contemporary,
Eudemus, shrewdly remarked that “they changed geometry into a literal science;
they diverted arithmetic from the service of commerce”.
The Joy of Finite Mathematics is written to show at a very basic level,
that mathematics is useful to virtually everyone, especially those students
Mathematics is the “brain” for
Engineering
Health Sciences
Education
Social Science
Physical Sciences
Economics
Finance
Environmental Sciences
Global Warming
Music
Art
Among others…
Discrete
Apart or detached from others; separate;
distinct
Absolute
Not mixed or adulterated; pure
Relative
Something having, or standing in, some
relation to something else
Continued
To go on with or persist in; to continue an
action
Stable
Not likely to fall or give way; firm, steady
Moving
To pass from one place or position to
another

who do not like mathematics as we approach mathematics as a language
used to describe simple and complex problems that we encounter in our
daily lives.
In the essay on “The Nature of Mathematics,” by Philip E B Jourdain, he
begins with “An eminent mathematician once remarked that he was never
satisfied with his knowledge of a mathematical theory until he could
explain it to the next man that he met in the street.” This is so very true
and we believe it is our responsibility in writing this text to explain to our
students the usefulness of mathematical methods and theories using real
world problems. Thus, the student has the right to ask “what is the usefulness
of mathematics?”
We have taken that aspect of the student asking such questions as our
responsibility in positively responding. We proceed to address this important
issue by raising several relative questions in the interdisciplinary structure of
mathematics that constitute the areas of the subject that we have identified as
“Finite Mathematics.” Thus, in what follows is the main thrust of the basic dia-
lects of mathematics for students whose primary interest is not the subject
matter, but how to enhance their understanding of the usefulness of mathe-
matics. For motivating the students we begin each branch of mathematics
by stating several real world questions, the answers to which will lead to
the importance and usefulness of mathematics. We believe that this interactive
approach will motivate the learning process and take our students on a very
“joyful ride” to learn finite mathematics.
1.2 WHAT IS LOGIC?
Logic is derived from the Greek λογικ
η meaning conforming to laws of rea-
soning. The branch of philosophy that treats forms of thinking, reasoning or
arguing is also referred to as Logic. Averroes defines logic as “the tool for
distinguishing between the true and the false.” Logic is divided into two
parts: inductive and deductive reasoning. Inductive reasoning draws con-
clusions based on specific examples whereas deductive reasoning draws
conclusions from definitions and axioms.
Thus, our goal in learning Logic is to be in a position to make logical deci-
sions regarding such questions as:
□ Politics: A politician claims “if you don’t vote for me, then you will not get
the tax cuts”—does this imply that if you do vote for him, that you will get
the tax cuts?
□ Health: If you work out more, then you will lose weight and tone your
muscles, and if you watch your calorie intake, you will lose weight. Does
this mean that if you lost weight that you must have both worked out and
watched your calorie intake?
□ Travel: If Athens is in Greece and Berlin is in Germany, then when I visit
Germany and not Athens, then does it follow that I went to Berlin?
□ Lottery: If Frank wins the lottery, then Frank will take you to dinner.
Frank did not win the lottery and did not take you to dinner. Did Frank lie?
□ Law: If you are 17, then you are a minor. Jordan is not 17; therefore, can
we conclude that Jordan is not a minor?
Intelligence is the ability to adapt to change.
Stephen Hawking
Archival Note
Averroes as he is known in Greek is Abū
’l-Walı̄d Muh
˙
ammad bin Ah
˙
mad bin Rushd,
and he defined logic and is the founder of
Algebra.
Debate between Averroes and Porphyry
Monfredo de Monte Imperiali Liber de herbis,
14th century
Number is the within of all things
Pythagoras
My goal is simple. It is a complete understanding of
the universe, why it is as it is and why it exists at all.
Stephen Hawking
1.2 What Is Logic? 5

It is the aim to teach logical reasoning to enable students to reason using
the art of deduction and to draw correct conclusions when confronted
with facts.
1.3 USEFULNESS OF SETS
The word “set” has more definitions than any other word in the English lan-
guage due to its many origins. One origin of the word set is from the Old
English settan meaning cause to sit, put in some place, fix firmly, and another
is from the Old French setta, meaning collection of things. The branch of
mathematics which deals with the study of sets is called Set Theory. The
modern study of Set Theory was begun by Georg Cantor and Richard
Dedekind in the 1870s. The language of sets can be used to define nearly
all mathematical “objects” such as functions.
Set Theory begins with a fundamental binary relationship (similar to logic)
between objects O and a set S, namely that of membership. Either an object
(element) belongs to a set, or it does not.
Usefulness of Sets: To present data, relevant information in a systematic
manner so that it will be visually attractive and easily understood and so that
it can be used effectively to address various questions of interest. Thus, our
goal in learning about sets is to be in a position to make categorical decisions
regarding such questions as:
□ Business: A store owner notes that more people like chocolate
muffins than blueberry muffins. With this information, how many
of each should be made? How many customers are expected to
purchase both?
□ Cancer: If survival is a function on the type of treatment(s) received,
then which treatment is better or is a combination of treatments better?
□ Meteorology: Given 20 readings of temperatures over a period of 14 days
taken at two relatively close stations, when comparing these temperatures;
do they appear to fall in the same temperature range?
It is our aim in learning Set Theory that students will be able to describe infor-
mation categorically as well as to be able to display information graphically in
Venn diagrams and use these graphics to support any inferences made
regarding relationships among the various sets.
1.4 COUNTING TECHNIQUES
Count is from the Old French counter meaning add up, but also tell a story.
Some of the first known use of counting was with shepherds who, when
tending their sheep, would tie knots in a rope as they sent their sheep out
to graze. In the evening, when the sheep returned to the fold for safety during
the night, the shepherd would untie a knot and if there were any knots in the
rope, they knew there were sheep that needed to be found. The branch of math-
ematics dealing with counting can extend from tally marks—making a mark
for each number and then tallying these marks, enumeration—counting aloud
Archival Note
Counting only involves the whole (counting)
numbers, 0, 1, 2, 3… It first started with the
natural counting numbers, and then we
introduced the number zero to represent
“nothing” or the number of elements in the
empty set. It was Jiu zhange suan-shu who first
used red rods to denote “positive” values and
black rods to denote “negative” values in his
writing Nine Chapters on the Mathematical Art.
For a long period of time, negative solutions to
problems where considered “false.”
Diophantus, in the third century AD,
referred to the idea of “2x+1050” as absurd.
Archival Note
Georg Ferdinand Ludwig Philipp Cantor was
a German mathematician, the inventor of Set
Theory, and the first to establish the importance
of one-to-one correspondence between sets.
Archival Note
Julius Wilhelm Richard Dedekind was another
German mathematician who worked with Cantor
and is also well known for his work in abstract
algebra.
Categorical
Unambiguously explicit and direct
Data consisting of nominal information
Qualitative data organized in a
contingency table
Venn Diagram
A set diagram that shows all possible
relations between finite collections

or on your fingers to more complex counting techniques such as combinations
and permutations.
Thus, our goal in Counting Techniques is to be in a position to count and
discern the implication of such questions as:
□ Social: At a social get-together of ten individuals, how many introductions
will be needed to ensure everyone has met face-to-face?
□ Coding: When coding a confidential letter using only the letters in
the English alphabet, how many distinct codings are there? How many
are needed such that no letter is mapped to itself in the coding?
□ Civics: A board of directors consisting of ten women and 15 men need
to form a five member committee to oversee next year’s fund raiser.
How many possible committees consist of exactly three men and
two women?
□ Job Assignment: A real estate agency has ten realtors and only nine
new property listings. How many possible assignments of realtors to
a house?
□ Diet: There is a list of ten fruits you are willing to eat and your goal is to eat
four fruits a day. To mix it up each day you create a meal plan that covers
all possible combinations of four out of ten fruits. How many options are
there for fruits in this meal plan?
The purpose in learning various counting techniques is to enable the student
to determine the logistics necessary in such detailed coordination of a complex
operation. This ranges from counting people or supplies to organizing com-
mittees and daily life.
1.5 PROBABILITY
Probability is from the French probabilite’ meaning quality of being probable
or something likely to be true. Probability is the branch of mathematics which
is a way of expressing knowledge (or belief) that an event will or will not occur
numerically, a form of empirical inductive reasoning leading to statistical
inferences.
The idea of probability originated with games of chance in the seven-
teenth century. The earliest writings in the area were the result of the collab-
oration of the eminent mathematicians Blaise Pascal and Pierre Fermat, and a
gambler, Chevalier de Mere. To them, there seem to be contradictions
between mathematical calculations and the events of actual games of chance
involving throwing dice, tossing a coin, spinning a roulette wheel, or
playing cards.
Thus, our goal in learning about Probability is to be in a position to compute
and interpret the relevance of probabilities and address such questions as:
□ Breast Cancer: A patient goes to the doctor with a lump in her breast.
What is the probability that it is a tumor? What is the probability that it
is cancerous?
□ Finance: What is the probability that the value of the Dollar will be higher
than the Euro in 2015?
□ Sports: What is the probability that the USF quarterback will complete
half of his passes in a given game?
□ Engineering: What is the probability that a computer software package
will fail?
Combination
When r out of n objects are taken without
replacement and without distinction in
ordering
Permutation
When r out of n objects are taken without
replacement and with distinction in
ordering
Counting is the religion of this generation. It is its
hope and its salvation.
Gertrude Stein
It’s not the voting that’s democracy, it’s the
counting.
Alfred Emanuel Smith
Innumerable actions are going on through us all the
time. If we started counting them, we should never
come to an end.
Vinoba Bhave
Music is the pleasure the human mind experiences
from counting without being aware that it is
counting.
Gottfried Leibniz
Probability is the very guide to life.
Cicero
Probability is expectation founded upon partial
knowledge.
George Boole
1.5 Probability 7

□ Sociology: What is the probability that a disadvantaged child in an urban
area will pass the Florida Comprehensive Assessment Test (FCAT)?
□ Meteorology: What is the probability that a hurricane will obtain hur-
ricane status category 3 or more?
□ Statistics: What is the probability that the mean number of accidents
during New Year’s Eve will exceed that of the previous year’s number
of accidents?
□ Physiology: What is the probability that an experimental animal
will convulse upon administration of a certain pharmacological agent?
□ Education: What is the probability that an individual’s score on an intel-
ligence test will show significant improvement following a refresher
course in verbal skills?
It is our aim to learn some of the very basic aspects of probability so that we not
only answer questions such as those given above, but also to understand the
role the subject plays in our daily lives. Learning probability is intended to
put the student in a position to apply probability to any area of study that they
are interested in: statistics, engineering, operations research, physics, med-
icine, business, economics, accounting, education, sociology, physiology,
agriculture, meteorology, linguistics and political science, among others,
and to use this information to make knowledgeable decisions.
1.6 BERNOULLI TRIALS
Trial is Anglo-French meaning act or process of testing. A Bernoulli trial is
an experiment whose outcome is random, but has one of only two possible out-
comes: success or failure. The discrete probability distribution that we use to
answer such questions, among others, is the binomial or Bernoulli proba-
bility distribution; a mathematical expression that generates the actual prob-
ability for specific inputs that relate to a given question. We encounter many
important situations that can be characterized by a discrete random variable
with this developed distribution.
It is our goal in studying Bernoulli trials to put ourselves in a position to
compute binomial probabilities and address such questions as:
□ Births: A baby born less than 36 weeks is consider premature. What is the
probability that a baby will be born premature?
□ Medicine: What is the probability that a given drug will be effective to
cure a specific disease?
□ Politics: What is the probability that Candidate A will be elected president
of the US?
□ Gambling: What is the probability that I will obtain an odd number in a
single roll of a fair die?
□ Computers: What is the probability that the computer you purchased
online will be operable (non-defective)?
We will learn how to use this very important probability distribution to answer
the above questions, among others.
1.7 THE BELL-SHAPED CURVE
The Bell-shaped Curve is commonly called the normal curve and
is mathematically referred to as the Gaussian probability distribution.
Discrete
A type of measure such that the outcomes
are separate and distinct
Random
Taken such that each individual is equally
likely to be selected
Variable
A distinct characteristic of an individual
to be observed or measured
How dare we speak of the laws of chance? Is not
chance the antithesis of all law?
Joseph Bertrand

Unlike Bernoulli trials which are based on discrete counts, the normal dis-
tribution is used to determine the probability of a continuous random
variable.
The normal or Gaussian probability distribution is the most popular
and important distribution because of its unique mathematical properties,
which facilitate its application to practically any physical problem in the
real world; if not for the data’s distribution directly, then in terms of the dis-
tribution associated with sampling. It constitutes the basis for the devel-
opment of many of the statistical methods that we will learn in the
following chapters.
Thus, our goal in studying the Bell-Shaped Curve is to put ourselves in a
position to compute and interpret probabilities associated with continuous
random variables and address such questions as:
□ Cancer: What is the probability that in a given group of lung cancer
patients, an individual selected at random is Asian?
□ Education: What is the probability that a student will have a final grade in
finite mathematics between 85 and 95?
□ Sports: What is the probability that a given lineman’s weight on the USF
football team will be between 275 and 325 pounds?
□ Rainfall: What is the probability that the average rainfall in the
State of Rhode Island in the year 2012 will be between 16 and 24
inches?
□ Chemistry: What is the probability that an acid solution made by a spe-
cific method will satisfactorily etch a tray?
The objective in learning the mathematical properties of the normal prob-
ability distribution is to realize its usefulness in characterizing the
behavior of continuous random variables that frequently occur in daily
experience.
1.8 STATISTICS
The branch of Statistics, meaning quantitative fact or statement, is becoming
more widely accepted as a necessity for understanding all aspects that
influence our daily lives. In almost every field of study, statistics is used to
estimate the unknown, a characteristic of the individual we would like to know
about in a given population. It is similar to the Scientific Method, in that we
must first understand and clearly state the problem, gather the relevant infor-
mation, formulate a hypothesis and test this hypotheses by recording and ana-
lyzing the data, before we can interpret the data and state our conclusion.
The basic idea behind descriptive statistics is to reduce a set of data down
to one piece of information that describes some aspect of the data—an estimate
of the population mean, or its central tendency, deviation, range, extremes, etc.
Thus, our goal in studying Statistics is to be able to analyze and interpret real
world data so that we will better understand the phenomenon that we are
studying and address such questions as:
□ Business: What is the mean profit made per hours of production time as a
function of employees on the floor?
□ Politics: What percentage of the people truly desire a tax increase given
only 40% of individuals vote?
□ Chemistry: What is the point of saturation for carbon dioxide in the
atmosphere?
□ Medicine: What is the mean tumor size in a patient with brain cancer?
Continuous
A type of measure such that the outcomes
are dense, that is, between any two
outcomes, other possible outcomes exist.
The graph of the normal probability
distribution is a “bell-shaped” curve, as shown
in the figure above. The constants μ and σ are the
parameters.
The area under the curve represents the
underlying probability of the situation.
Statistics
The art of decision making in the presence
of uncertainty
As opposed to statistic—
a numerical datum
Hypothesis
Greek meaning “to suppose”
A statistical analysis, properly conducted, is a
delicate dissection of uncertainties, a surgery of
suppositions.
M.J. Moroney
Statistics may be defined as “a body of methods for
making wise decisions in the face of uncertainty.”
W.A. Wallis
1.8 Statistics 9

□ Engineering: What is the mean maximum load (kN) for a fishing line?
What is the mean elongation?
□ Astronomy: What is the mean temperature fluctuation in the Sea of Tran-
quility on the moon?
□ Agriculture: What is the mean yield of corn per acre given the number of
acres planted?
□ Sociology: What is the mean number of texts sent by a cellular phone
user in a given month? Is there a difference in usage between teens
and adults?
The point in learning basic statistics is to be able to efficiently gather,
organize, analyze and interpret data in order to address questions that arise
from every field of study and that apply to everyday living in a growing global
society.
1.9 GEOMETRY
Geometry is from the Ancient Greek word γεωμετρια meaning measurement
of earth or land. This branch of mathematics is concerned with questions
regarding the shape, size, relative positions and properties of space.
Euclidean geometry is a mathematical system that assumes a small set of
axioms and deductive propositions and theorems that can be used to make
accurate measurement of unknown values based on their geometric relation
to known measures.
Thus, our goal in studying Geometry is to be able to accurately measure
the world around us, perform basic calculations that address such
questions as:
□ Agriculture: Using similar triangles, given the height of a stick and the
length of its shadow at 2:00 PM, measuring the shadow of the tree at
the same time, determine the height of the tree.
□ Carpentry: If two boards, mitered at a 60° angle, are reversed and
attached to create a frame, what is the angle formed by the joint?
□ Playground: How large should a sandbox be if only 5 ft of wood is
available and how much sand is needed if the wood is 6 in. tall?
□ Rubik Cube: How many squares are there on the surface of a Rubik Cube?
□ Business: If a showroom has 10,000 square feet of space to be converted
into offices, but must leave 5000 square feet for the showroom floor and
each office must be 200 square feet of space, how many offices can be
created at most?
□ Chemistry: What is the shape of a sugar molecule? How does this differ
between mono-dextrose and poly-dextrose sugars? What is the difference
in volume?
The intention behind learning Geometry is to enable the student to be pro-
ficient in both the art and science of geometry. Geometry is used in areas
ranging from graphic design to Einstein’s theory of general relativity; when
a surveyor plots land, a manufacturer determines the best packaging for a stack
of spherical oranges to be shipped or a car manufacturer redesigns a parabolic
headlight, for example.
1.10 ARITHMETIC AND ALGEBRA
Arithmetic means the art of counting, and Algebra means reunion of broken
pieces. Arithmetic is the oldest and most elementary branch of mathematics
and deals with the study of quantity such as those that result from combining
other quantities, which leads directly to Algebra. Algebra is a branch of
Euclid is the Father of Geometry best known
for his book Elements which consist of 13 books
covering Euclidian Geometry.
Title page of Sir Henry Billingsley’s first English
version of Euclid’s Elements, 1570
I’ve always been passionate about geometry and the
study of three-dimensional forms.
Erno Rubik
There is geometry in the humming of the strings;
there is music in the spacing of the spheres.
Pythagoras
Give me a lever long enough and a fulcrum on which
to place it and I shall move the world.
Archimedes

mathematics outlining arithmetics, the rules of operations such as addition,
subtraction, multiplication and division, but also relations such as equal-
ities, inequalities and functions.
Arithmetic and Algebra are the building blocks of most areas in math-
ematics, usually taught as part of the curriculum in primary and secondary
education. However, even at university level, these topics are extremely
useful allowing general formulations to be the first step in the systematic
exploration of more complex problems that can be solved using Math.
Thus, our goal in studying Arithmetic Algebra is to be able answer such
questions as:
□ Health: Based on the nutritional information for three dog foods
based on three required nutrients, how much of each type of dog
food should be included in a single serving to optimize the nutritional
intake?
□ Farming: Given 100 feet of fencing, how should the length of a pen be
related to the width, if the fence is to create two adjacent pens sharing
a common size with maximum area?
□ Business: If you sell tickets for $20 each and you sell as many as you can,
which beforehand is an unknown quantity, x, how does your profit relate to
this unknown value x?
□ Social: If you know that you have x adults coming for dinner and one
child, and each adult eats three manicotti shells and the child eats one,
how many shells must be made, y, as a function of the number of adults
invited, x.
The aim of learning Arithmetic and Algebra is to refresh the student’s
understanding of the subject matter and to introduce more relevant uses
of this dialect of Math. Remember: we think faster than we speak and
we speak faster than we write. Therefore, to address large complex
problems such as building a bridge, we need a very short handed language.
This universal language is Math, and Arithmetic and Algebra are a large
part of this language.
1.11 FINANCE
Finance means to ransom, or to manage money. This science of funds man-
agement includes business finance, personal finance and public finance. Our
goal is to use mathematics to teach the student to have a better understanding of
basic personal finance; such finances will include savings and loans in terms
of time, money, risk and how they are interrelated in addition to spending and
budget.
Thus, our goal in studying the Basics of Finance is to be able to understand
and manage personal finances and address such questions as:
□ Personal Budget: How much do you spend each month on Rent, Elec-
tricity, Phone, Internet, Food, Gas, Insurance, etc.
□ Wedding: How much can I afford to spend? If I finance a wedding on
credit, how long will it take to pay off this debt and how much will it
eventually cost?
□ Transportation: What should be the maximum payment I should agree to
in order to ensure my vehicle is not repossessed.
□ Housing: Can you afford to move out of your apartment and into a house?
What is the expected down payment? Inspection fees? The expected
property taxes?
Diophantus is traditionally known as the Father
of Algebra, but this has recently put up to debate
in that Al-Jabr, the author of Arithmetic gives
the elementary algebra before Diophantus in
200-214 CE.
How does y relate to x?
In the business world, the rearview mirror is always
clearer than the windshield.
Warren Buffett
1.11 Finance 11

□ Home Repairs: How long would it take to save enough to replace an air
conditioning unit? How much can be saved be investing into a sinking
fund versus using credit?
□ Credit: If you make the minimum required payment and have a
minimum purchase each month, how long will you be indebted to the
creditor?
□ Christmas Funds: How much needs to be put into a sinking fund in order
for you to have saved up $1000 in a Christmas fund over a period of
11 months starting in January?
In is imperative in today’s economy that everyone has a basic under-
standing of finance. Many individuals are overwhelmed when confronted with
mounting bills or credit; however, it is important that they budget, even if the
final amount is negative. Once we are aware of the problem, we can begin to
work out the solution. Understanding the basic mathematics behind Basic
Finance, we will be in a position to make positive changes in our present
financial state and better plan for the future.
1.12 GAME THEORY
Game Theory is a study of strategic decision making between two rational
decision-makers. Here, we address two-person zero-sum games; games
designed such that one players gain equals the second player’s loss.
□ Strictly Determined Games: The Saddle Point
□ Games with Mixed Strategies
□ Reducing Matrix Games to a System of Linear Equations
CRITICAL THINKING AND BASIC EXERCISE
1.1. Who was the first to call himself a philosopher?
1.2. What is Pythagoreanism?
1.3. What does the word “matheno” mean?
1.4. Mathematics can be divided into what two branches?
1.5. Who wrote “The Nature of Mathematics”?
1.6. What word is derived from the Greek meaning conforming to laws of reason?
1.7. Distinguish between the two types of reasoning.
1.8. In Logic, which type of reasoning is used to draw correct conclusions when confronted with facts?
1.9. What does the word “setta” mean?
1.10. The modern study of sets began with two mathematicians; name them and state where they are from.
1.11. Sets are used for what type of measure: numerical or categorical?
1.12. What diagram is used to graph categorical information and to support any inferences made regarding relationships
among the various sets?
1.13. What are some of the first known uses of counting?
1.14. Name three counting techniques.
1.15. Name the area of study that is a form of empirical inductive reasoning leading to statistical inferences.
1.16. In what area of study are Blaise Pascal, Pierre Fermat and Chevalier de Mere known to have collaborated?
1.17. Who said “Probability is expectation founded upon partial knowledge”?
1.18. A binomial experiment is also known by what other name?
1.19. In a Bernoulli trial, there are exactly how many possible outcomes?
1.20. The binomial probability distribution is characterized by what type of random variable? Continuous or Discrete.
1.21. The normal probability distribution is also known by what other name?
1.22. Outline the steps associated with the Scientific Method.
1.23. Distinguish between Descriptive and Inferential Statistics.
If thou dost play with him at any game,
Thou art sure to lose, and, of that natural luck,
He beats thee ’gainst the odds.
Shakespeare
Budget
An estimate, often itemized, of expected
income and expense for a given period of
time in the future
A sum of money set aside for a specific
purpose
In the absence of the gold standard, there is no way
to protect savings from confiscation through
inflation. There is no safe store of value.
Alan Greenspan

1.24. List the points you need to learn in basic statistics to be an efficient researcher.
1.25. What branch of mathematics is concerned with questions regarding shape, size, relative positions and properties
of space?
1.26. Name the mathematical system that assumes a small set of axioms and deductive propositions and theorems that can
be used to make accurate measurement,
1.27. The art of counting is better known by what name?
1.28. Name the rules of operations in Algebra.
1.29. Why are Arithmetic and Algebra important?
1.30. Name the science of funds management.
1.31. Name the study of strategic decision making between two rational decision-makers.
SUMMARY OF IMPORTANT CONCEPTS
The first chapter, The Usefulness of Mathematics, introduces Mathematics and its history. This motivational chapter
answers the question “what is logic”; outlining the usefulness of Sets; the start of Counting Techniques and how counts
are the foundation of empirical probabilities. The first chapter also includes the usefulness of Mathematics in basic Prob-
ability and Statistics, Geometry, and Finance; an overview of basic Arithmetic and Algebra along with Game Theory.
The second chapter on Logic covers statements and their truth values; outlines the symbolisms used to express state-
ments in the short hand language of Math including logical operators: conjunction, disjunction, negation and implication;
how to construct truth tables and determine equivalent statements. This chapter helps the student understand logical
reasoning by interpreting logical symbolism by giving their English translation. Properties of Logic covered include
Tautologies, Self-Contradictions, Paradox, Equivalence, and Algebra of Statements; Variations on the Conditional
Statement; Quantified Statements; Testing the Validity of an Argument and Applications of Logic.
The third chapter on Sets gives an introduction to Set Theory, covers collections of objects, the symbolisms used to
express these collections (sets) in the short hand language of Math including set operators: intersection, union, complement
and subset; and how they relate to logical operators. This chapter covers the Algebra of Sets, some basic counting principles
applied to sets.
The fourth chapter on Counting Techniques introduces counting principles beyond that of simple sets to that of the
Multiplication Principle, Permutations and Combinations, Distinct Orderings, and other counting techniques such as the
Binomial Theorem, and Pascal’s Triangle.
The fifth chapter on Basic Probabilities gives an introduction to probability and various definitions: personal probability,
empirical and theoretical. This chapter covers the experimental probabilities using sample spaces, the basic laws of prob-
ability, conditional probability and Bayes rule.
The sixth chapter on Binomial Probability introduces discrete random variables, discrete probability distribution in
general including expected value and variance followed by the Binomial Probability Distribution and the expected value
and variance for the Binomial random variable.
The seventh chapter on Normal Probability introduces continuous random variables and the Normal probability distri-
bution. This chapter also ties back in with discrete random variables covering Normal Approximation to the Binomial.
The eighth chapter on Descriptive Statistics covers gathering and organizing data; graphical representations of quali-
tative information and quantitative information; and measuring central tendencies and deviations from the center.
The ninth chapter on Geometry covers rounding and types of measurement; properties of lines: linear, linear pairs, two
lines and three lines; properties of angles: categorization and additive principles; properties of triangles: categorization and
similar/equivalent triangles; and properties of quadrilaterals and polygons. This chapter also covers area, surface area and
volume.
The tenth chapter on Arithmetic and Basic Algebra covers the real number system, basic arithmetic: addition, sub-
traction, multiplication and division; pattern recognition: sequences and series; algebraic expressions and relationships;
equations: equalities and systems of equations; and functions: linear and quadratic equation.
The eleventh chapter on Finance covers basic financing including sinking funds and amortization: various savings sit-
uations and comparison shopping: credit versus cash, leasing versus purchasing, and renting versus owning. This chapter
also covers effective rates and uses them to compare CD versus credit and comparisons of credit cards. There is also a section
on personal finance: how to create a monthly budget; insurance: what every homeowner should know and your credit report.
The twelfth chapter on Game Theory covers two-person zero-sum games. This chapter covers the Matrix Game; strictly
determined games, games with mixed strategies and instructions on how to reduce Matrix Games to Systems of Equations.

Chapter 2 Logic
2.1 Logic 16
2.2 Statements and Their Truth Values 17
2.3 Truth Tables 27
2.4 Properties of Logic 34
2.5 Variations of the Conditional Statement 41
2.6 Quantified Statement 44
2.7 Negating Statements 48
2.8 Testing the Validity of an Argument 54
2.9 Applications of Logic 66
Critical Thinking and Basic Exercise 76
Review Test 82
References 85
Number is the within of all things.
PYTHAGORAS
Truth Belief
Knowledge
Propositions
Copyright © 2016 Elsevier Inc. All rights reserved. 15

In the field of mathematics, Aristotle is probably best noted for his contributions to the methods of proofs. He was the first to
provide clear distinctions between axioms, postulates, and definitions. In addition, he contributed theorems of geometry,
infinity, and continuity. He was considered to be a philosopher, but the philosophy of his time included what is now classified
as natural sciences. Among Aristotle’s writings on logic (called later the Organon) are Prior Analytics, Posterior Analytics,
and Sophisms. In these and in his other works, he systematized the formal rules of logic and introduced syllogism, a form of
deductive reasoning.
Aristotle was a student of Plato’s Academy and later became a teacher. When he was forty-one, he began to supervise the
education of Alexander the Great, for which he received the beginnings of his fortune. He later taught in the Lyceum in
Athens and began amassing a book and map collection for a museum of natural history. This arrangement eventually
led to a “school” after his death at the age of sixty-two.
Aristotle’s influence was so encompassing and pervasive that many of his contributions were not even questioned until
the middle of the nineteenth century even though many of his theories were incorrect. His writings, including accurate as
well as misdirected ideas, were accepted as the ultimate authority during the medieval period and were upheld by the Roman
Catholic Church beyond the time of Galileo (Aristotle had believed in the geocentric; i.e., earth-centered solar system,
which Galileo unsuccessfully argued against in the Inquisition). Although his dogmatic followers deterred further advances
for many centuries, Aristotle did much to advance science in his time. His many fields of study included biology (he devised
classifications for all kinds of plants and animals), metaphysics and logic, ethics and politics, rhetoric and poetics, weather,
and the other physical sciences.
2.1 LOGIC
Why should we have chosen to begin the study of finite mathematics with a
chapter on logic? The following argument is offered by way of illustration:
Mathematics must be based on logic. This is a basic course in mathematics
with emphasis on its usefulness. Therefore this course must be based on logic.
An argument of this form is known as a syllogism. A syllogism is a typical
Aristotelian argument. Aristotle gave the first systematic treatment of the
principles of logical reasoning which earlier Greeks had begun to formulate.
Aristotelian logic is the fundamental form of logical reasoning which is still
utilized today.
The assertion that mathematics must be based on logic is justifiable
because virtually all mathematical results are obtained by logical
deductions from other previously obtained results now generally accep-
ted as true, or from assertions which have been assumed without
proof. Sometimes, making logical deductions is not as straightforward
and simple as we might like; thus, it is true here as in many other situa-
tions that possession of a set of rather specific rules makes the task
much easier.
George Boole (1815-1864), an English mathematician, integrated
logic into algebra and essentially founded the field of mathematical logic.
He introduced the use of symbols to represent statements or assertions,
which greatly increases the ease and speed of manipulation of concepts
in deductive logic. Mathematical logic is also known as symbolic
logic for this reason. In this chapter some of the fundamental concepts of
mathematical logic will be discussed with a view toward enabling the
reader:
Logic

Derived from the Greek word logos
which means reason or discourse
Syllogism

An argument supported by two
premises; deductive reasoning

An extremely subtle, but sophisticated
argument
Aristotelian

Of, pertaining to, based on, or derived
from Aristotle or his theories.
1. To be able to apply logic to analyze
problems, and
2. To obtain proficiency in the correct
methods of logical reasoning.
16 Chapter 2 Logic

… no general method for the solution of questions in the theory of probabilities can
be established which does not explicitly recognize … those universal laws of
thought which are the basis of all reasoning …
George Boole
To this end, we must consider that we think faster than we speak,
we speak faster than we write, therefore to think quickly and communi-
cate these ideas, we must learn to abbreviate almost everything. A summary
of modern symbolic logic can be found in the summary, at the end of the
chapter.
2.2 STATEMENTS AND THEIR TRUTH VALUES
In this section we shall discuss one of the basic concepts of logic; namely, that
of a statement. We shall also introduce some other important terms and
symbols. We begin with a definition.
Definition 2.2.1 Statement
A statement is a declarative sentence which is either true or false, but not both. We shall
denote statements symbolically by lower case letters p, q, r…
We judge statements with respect to their truth value. That is,
Definition 2.2.2 Truth Value
The truth value of a statement is the truth or falsity of the statement. We shall denote
true by T and false by F.
Example 2.2.1 Classify Sentences
Consider the following sentences; classify each as statement, question or command:
(a) London is in France
(b) 3 + 55 8
(c) Who is here?
(d) Put the book on the shelf.
(e) Sometimes it rains.
Solution
Sentence (a) is a false statement, and sentences (b) and (e) are true statements. However,
sentences (c) and (d) are not statements because neither can be assigned a truth value of
true or false. Sentence (c) is a question and sentence (d) is a command.
True Statements:
▪ Monday is a day of the week
▪ 5 is a natural number
False Statements:
▪ January is a day of the week
▪ 5 is a negative integer
Facts:
(a) London is, in fact, not in France
(b) 3 + 5 is equal to 8
(c) Not a statement
(d) Not a statement
(e) Sometimes it does rain.
Simple Statements:
▪ Today is Monday
▪ Tomorrow is Christmas
▪ The sun is shining
▪ There are rain clouds in the sky
▪ I will study English
▪ I will study Math
2.2 Statements and Their Truth Values 17

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Yes, it was indeed Edward Temple, upon whom she gazed with ill-
defined ideas—and feelings of bewilderment and perplexity—her
high-wrought expectations unable all at once to sink themselves to
the level of natural composure—pale, agitated, and trembling,
without further greeting or explanation,
She showed the ring.
I found it, she said with almost hysterical incoherency, and
thought perhaps—but your's it cannot be—and yet it is strange—the
initials are the same—but—can it really be, that your crest—your
arms also are similar?
For all reply he gently took the ring from her outstretched hand, and
in silence seemed to examine it. Then without looking up, and in a
low, calm voice he said:
You expected I conclude, to find the owner had been Eugene
Trevor?
No, not Eugene, Mary quietly replied, restored to greater self-
possession, but perhaps, I thought—it was a random idea—that
perhaps it might have been his brother Eustace.
The ring dropped suddenly from her listener's fingers, as she uttered
these last words.
And what, he murmured, having stooped to raise it from the
ground, and what interest can Miss Seaham take in that ill-starred,
that unhappy man; that outcast, alien brother, that her mistake
should cause disappointment, such as I so plainly perceive it to have
occasioned her?
Mary probably attributed to wounded feeling the trembling pathos of
the speaker's voice, for with all the simple earnestness of her kindly
nature, she hastened in gentle soothing accents to reply:

Mr. Temple—if disappointment was the first impulse of my feelings
—believe me, when I say, there is scarcely any one else, with a
weary sigh, the tears gathering in her eyes, with whom a meeting
so unexpected, could just now have afforded me such unmixed
pleasure.
For one short moment her hand was retained by the so-called Mr.
Temple in a trembling pressure, which appeared to speak all his
heart's grateful acknowledgement, whilst those dark eyes fixed
themselves upon her face with mournful earnestness of expression.
But the next moment, with a low-breathed sigh, which might have
seemed the echo of her own, he released her hand, and turned
away his head.
You are kind to say this, he murmured, for myself, I can only
declare this meeting to be a happiness such as I had hardly
expected ever to taste again in this world. But, he anxiously
inquired, will you again permit me to inquire the reason of the more
than common—nay even, taking into consideration his relationship—
more than natural interest, it would appear you feel in the
unfortunate Eustace Trevor.
The earnest melancholy of his tone thrilled on Mary's heart.
Mr. Temple, she said eagerly, you speak with feeling on this
subject, can it, oh! can it be possible that you have ever seen, ever
known Eugene Trevor's brother? Oh, tell me if this is really the case,
for you say true—in more than common degree—quite
independently of selfish motives, connected with my own happiness
—has my interest been excited in his discovery. It has been most
strongly awakened in the fate, and history of one who has lately
been brought before me in a light so charming yet so sad. Oh! Mr.
Temple, you do not deny the fact. Then, tell me, only tell me where
he can be found?
Eustace Trevor had turned upon her the full light of his radiant
countenance, radiant with a new and strange delight, the nature of

which she could not comprehend; but as, with clasped hands and
beseeching countenance, she uttered this latter inquiry, it was
answered by a gesture, seeming to imply by her listener ignorance in
the required information.
You, then, did not know him? she resumed, with renewed
disappointment in her tone.
I did know him—ah, too well! was the murmured reply, his eyes,
with a strange and mysterious expression, fixed upon the ground.
Very pale suddenly grew Mary's cheek as she looked upon him thus.
Her lips parted, and her heart beat fast as from the shock of a
strange and sudden idea, which flashed across her senses. But she
put by the suggestion as the wild improbable coinage of her own
high wrought imagination. She remembered too what had struck her
often vaguely before, and also her brother's remark on a former
occasion, with reference to the same resemblance. But when she
looked again, the glowing illusion had faded, her companion was
again calmly regarding her, again asking—in what she esteemed a
cold and careless tone of voice—from whom it was, she had received
the impression respecting Eustace Trevor, to which she had just
alluded.
It was his friend, and my cousin—Louis de Burgh, who first spoke of
him to me in such warm and glowing terms; but he chiefly raised my
interest by the beautiful but melancholy picture he drew of his
devoted affection for his mother—that mother, she added in a low,
sad tone, with whose unhappy history, I then for the first time was
made acquainted—indeed it caused his very affliction to become
almost holy in my eyes—by showing it to have been but the crisis of
his high and sacred grief. Mr. Temple, she continued with
enthusiasm; there seems to me something, if I may so speak,
almost God-like in the pure and devoted love of a strong proud-
hearted man towards his mother; and it is God-like, for was not the
last earthly thought—the last earthly care of Him who hung upon the
cross, even in his mortal agony—for his mother!

The speaker's glistening eyes were raised above or she might have
seen tears indeed,
Such as would not stain an angel's cheek,
also irradiating the eyes of that strong proud-hearted man, as she
so expressed herself—who was standing by her side.
But she could not have heard—for it was not breathed for mortal ear,
the deep and fervent cry: My Mother! which her innocent words,
like thrilling music by the winds, struck from the secret chords of
that manly tender heart.
But this was a theme Eustace Trevor's melting soul could not trust
itself to pursue; not indeed, without it were first allowed him to cast
away all subterfuge and disguise, and at the feet of that good, kind,
and gentle girl, open his whole bruised and desolate heart, to
receive that Heavenly balsam of pity and consolation, she had ready
stored within her breast for the faithful son of that wronged and
sainted mother!
And could this be done? Had he not for the sake of this same gentle
being, in some sort pledged himself to such an extent, that yielding
to the impulse would be baseness and dishonour.
Alas! as in all divergement from the direct and natural paths of
human action, in whatsoever spirit they may have been entered
upon, the time must come—circumstances must arise—when the line
of duty becomes bewilderingly shadowy and indistinct, even to the
most conscientious and true-hearted.
How few can steer their way unwavering through the straightened
pathway of a false position. It is not there, that like a stately ship he
can vigorously part the waves of circumstance or temptation,
And bear his course aright.
Nor ought for tempest doth from it depart,
Nor ought for fairer weather's false delight.

Therefore, with an effort over his feelings which might have made
him appear unaffected by the sentiments his companion had so
touchingly expressed, he was forced merely to reply: Yes, Louis de
Burgh was his friend; and it would be very gratifying to Eustace
Trevor to know that one friend at least in that world he has
abandoned, retains him in such affectionate remembrance. And his
brother—he added, with more hesitating restraint in his tone, did
you never receive anything of the same impression from him?
Eugene, Mary answered with some slight embarrassment, rarely
ever enlarged upon a theme which of course had become connected
in his mind with painful feelings.
Painful indeed! was the other's significant rejoinder.
Never but once, Mary continued, did I venture to question him
upon the subject with any minuteness, and then he manifested such
strong and painful emotion that I never afterwards approached it
willingly. But at that time, she added with a sigh, I had certainly
heard very little of his brother, but the dark and terrible malady with
which he was afflicted. Mr. Temple, she continued anxiously, is not
his complete disappearance most mysterious and inexplicable? and
does it not appear to you almost impossible, that all the means
which have been taken for his recovery could have been so
completely unattended by success, supposing he were still alive?
But have any such means been taken? her companion asked with
some marked curiosity.
Oh yes! she hastened to reply on Eugene's part at least.
A peculiar smile played on her companion's lips. It did not fail to
strike Mary, and the incredulity it seemed to imply caused her
feelings now so peculiarly sensitive upon that point, to be
immediately up in arms.
Mr. Temple, can you for a moment doubt this fact, he is Eugene's
own brother, and— she added in a low voice, the crimson blood at

the same time mantling her cheeks, as the remembrance that she
was addressing a rejected lover, pressed more consciously upon her,
he had interests of a different nature, closely connected with the
assurance of his lost brother's fate?
Mr. Temple started with sudden excitement.
Indeed! he exclaimed, then averting his head, he added, as if the
utterance of each syllable was a separate pang. Do you mean to
say that there is still a question of this marriage?
There is, she replied; though of a very remote and undefined
nature, our engagement still subsists.
Having said this with no little embarrassment of manner, the same
feeling probably caused her to raise her arm from the fountain, over
which she had been unconsciously leaning, and by tacit consent they
turned away from the spot, silently beginning to retrace their steps.
They had not proceeded thus many yards, when Arthur Seaham
appeared in sight, accompanied by a second person, who Mary, with
an exclamation of delighted surprise, recognized as Mr. Wynne,
concerning whom in the absorbing interest of the last hour she had
no time to seek information.
The good clergyman on his part, who had fallen in with her brother
at the hotel, was charmed beyond expression by this fortunate and
unexpected meeting with his own dear children, (so he called Mary
and Arthur;) and peculiar was the glance of interest which beamed
from his kindly eyes, as having gazed anxiously into Mary's face, he
turned then towards her companion, who nevertheless with his fine
countenance only a little paler than usual, was exchanging kind and
cordial greetings with young Seaham.
Oh! Mary, Mary! the good clergyman whispered, as he drew his fair
friend's arm within his own and walked on, the others following
together behind, I have heard sad stories of you, little quiet one,
since I saw you last;—trampling noble flowers under your feet, and
grasping at thorns, which something in that sweet face of your's tells

me have not failed to do their wounding work. This comes of reading
all that dreamy poetry I used to warn you against. A good and
pleasant thing it is in its degree, but too much of it dazzles and
deludes the senses, till at length they come to be unable to discern
darkness from light, good from evil. Well! well! he added, as Mary
pretty well accustomed by this time to indirect attacks of this nature,
attempted no defence, but with a faint melancholy smile, only
drooped her head in silence and resignation. Ah! well, even now
who knows! The Almighty never will permit his little ones to walk on
long in darkness, but in the end ever leads them by secret ways into
safe and quiet pastures.

CHAPTER VI.
The stern
Have deeper thoughts than your dull eyes
discern,
And when they love, your smilers guess
not how
Beats the strong heart, though less their
lips avow.
BYRON.
The victory is most sure
For him, who, seeking faith by virtue,
strives
To yield entire submission to the law
Of conscience.
WORDSWORTH.
Arthur, this can scarcely be possible, Mary exclaimed with almost
trembling solicitude, when alone with her brother, he informed her of
the proposal Mr. Wynne had made—and he had unhesitatingly
accepted—that he and his friend Mr. Temple should join their party
during the succeeding week's tour.
Not if it is disagreeable to you, Mary, certainly, was the brother's
reply; otherwise I must say I can see no objection to the plan; nor
does Mr. Wynne either it seems, as he made the proposal, being of
course aware by this time of the past circumstances respecting you
and Temple. All that of course is an affair over and forgotten,

particularly when made aware how matters stand with regard to
your engagement with Trevor; so on your part, you will have nothing
to fear. It only rests with him, I should think, to determine whether
he is equal to the ordeal of your society, though to judge by his
countenance just now, firm and calm as a statue, after a meeting
which must have put his feelings rather to the test, I should say
there was not much doubt upon the matter.
'Nay, if she loves me not, I care not for her.
Shall I look pale because the maiden blooms,
Or sigh because she smiles—or sighs for others.'
No—no, Miss Mary, that is not our way, however it may be with you
ladies in cases of the kind.
'Great or good, or kind, or fair,
I will ne'er the more despair;
If she love me, this believe,
I will die e'er she shall grieve,
'Be she with that goodness blest,
Which may merit name of best.
If she be not such to me—
What care I how good she be.'
Thus the brother playfully sung and quoted, though whether the
philosophical doctrine the old poet implied in his song had the effect
of easing his listener's mind upon the point in question, her faint and
absent smile was not exactly calculated to declare; though perhaps
could he have read aright the secret history of that anxious
countenance, he might have seen how far less any such
considerations were agitating his sister's mind than the
remembrance of Eugene's strange and angry excitement in the
Edinburgh gardens, on the subject of this same Edward Temple; and
the question now chiefly agitating her breast to be, whether she
could without treason to her lover, place herself in the position and

circumstances now under discussion—yet what was she to do? She
knew that Arthur could not enter into her feelings on this point;
besides, was there not some unconfessed leaning in her secret heart
in favour of the arrangement. For that interview of the morning, and
the circumstances from which it took its rise; had it not aroused
ideas of perplexity, interest, and anxiety in her mind? was there not
still much left unaccounted for and unexplained?
She mentioned the ring to her brother. He was surprised, and
thought it a strange coincidence, though certainly it did often
happen that families of different names, bore the same crests,
sometimes the same arms.
Mary's recognition of the impression showed at least there to be,
some connection between Eugene Trevor and Mr. Temple. Arthur
could easily gain explanation from Mr. Wynne on the subject. He also
was often puzzled to know to what family of Temple his friend
belonged.
But, before time or explanation was given for any such inquiry, the
little party yielding themselves passively as it were to the irresistible
force of circumstances which had so singularly united them, were
pursuing their way over the enchanted ground Arthur had previously
marked out for their excursion, most of which the two more
experienced travellers had already explored, but gladly retrode for
the benefit of their young companions.
By sweet Val d'Arno's tinted hills,
In Vallambrosa's convent gloom,
Mid Terni's vale of singing rills,
By deathless lairs in solemn Rome.
* * * * *
Ruin, and fane, and waterfall.
They wandered delightedly, and never did Mr. Wynne and Arthur
cease to congratulate themselves and one another; the latter, on the
valuable acquisition he and his sister had gained in such able

cicerones as himself and his companion; whilst Mary and Mr. Temple,
by their silence only, gave testimony to the same effect.
Yes, it were well for the good Mr. Wynne and the young and hopeful-
hearted Arthur
Cheerful old age, and youth serene,
to yield themselves to the charm of sunny skies and classic ground,
and to feel almost as if earth wanted no more to make it Heaven.
A calm and lovely paradise
Is Italy for hearts at ease.
But for the other two, as may be supposed, there wanted something
more, or rather something less, to render their enjoyment as full and
unalloyed.
For in spite of all Arthur had urged to the contrary, it was too plainly
evident that something there was—a restraint—a consciousness,
influencing their secret feelings, and imparting themselves to their
outward demeanour, in common intercourse one with another; which
no exciting or absorbing diversities of scene or circumstance could
entirely dissipate or dispel.
Sometimes indeed, Mary, carried away by the delight of the
moment, would forget whose eye had fixed itself for a brief moment,
with such earnest interest, on her countenance; or even meet
unshrinkingly the glance, the smile of sympathy, which her
murmurings of enraptured admiration at times drew forth.
Sometimes unconsciously, as if it had been only as a portion of the
magic spell which hung on all around her, she found herself listening
to that voice, whose few, calm, graphic words had power to throw
desired light on some old haunt or story—or touch with a bright glow
the scene before them, or oftener turn away with a startled look of
anxious thought as if some sudden association or remembrance
recalled her to consciousness, and broke the spell.

Too happy to be your guide and guardian, through scenes and
beauty which even your lively imagination is incompetent to
conceive!
Did the words, which had once proceeded from those same lips,
thrill upon her recollection? or was it only the jealous disapproval of
her lover Eugene which would start up to trouble her on such
occasions?
Whilst Eustace—it would be vain to tell what caused the quick
transition of that glance or smile into the cold and rigidly averted
brow, or caused to die away upon his lips words whose inspiration
sprang from a source which could not be worthily encouraged.
Thus, day after day went on, and brought but diminished
opportunity of touching on those points of interest so near her heart,
and concerning which she more and more became possessed with
the vague and restless fancy, that Mr. Temple possessed more power
than any one imagined of enlightenment; for she avoided, as much
as possible, finding herself alone with him, and if at times, as
inevitably it occurred, they were thrown together apart from the
other two, Mary's haunting vision of Eugene's jealous disapproval of
her intimacy with Mr. Temple would cast a restraint over her feelings,
and made her shrink from availing herself of the favourable
opportunity thus afforded.
Of course Mr. Wynne—and through him Eustace Trevor had soon
learnt from Arthur every particular relating to his sister's situation
with regard to Eugene, and the effect produced upon the latter by
the circumstances which transpired, was evidenced only by the calm,
rigid expression which settled on his interesting countenance—only
subdued into soft and gentle melancholy, when at times, unobserved
by herself, his eyes could fix themselves on Mary; and as for meeting
her half-way, in any renewal of the subject, so particularly discussed
near the fountain that first morning of their meeting, he, with almost
equal pointedness, might have seemed to avoid any occasion which
could tend to its revival.

On the other hand, from Mr. Wynne the more unconscious and
unsuspecting Arthur could gain little satisfactorily information on the
topic on which he had promised to make inquiries. He always fought
off any cross questioning on any particular subject connected with
his friend Temple.
Indeed this was easy enough to do; for heart and soul absorbed in
the exciting enjoyment of scenes and circumstances in which he
entered with such enthusiastic delight, Arthur was not very capable
of pressing hard just now upon any serious point, not immediately
connected with the interest of the day or the hour.
But when Mary, with whom the old man had hitherto as skilfully
warded off any timid attempts on her part to draw him forth on the
subject on which he was vowed to secresy—when she, one sultry
afternoon, had been conversing for some time so delightfully with
her dear old friend, concerning days gone by, in the cool marble sala
of an old palazzo near Genoa, where they had found temporary
accommodation—without any preparation, fixed her earnest eyes
upon her companion's face, and said beseechingly:
Mr. Wynne, will you answer me one question? you are acquainted I
know, with everything concerning Mr. Temple; but I only wish to
ascertain one point; was he ever acquainted with Eugene Trevor?
The good man was taken by surprise, and displayed by his
countenance considerable signs of embarrassment, succeeded,
however, by equal symptoms of relief, when looking up he beheld
Mr. Temple, who had joined them unobserved, and must inevitably
have overheard Mary's words, and witnessed the perplexity they had
occasioned her friend.
Mary's cheek also flushed deeply; yet when the next moment Mr.
Wynne, with some careless excuse for leaving them, had walked
away, and she found herself alone with him who best could answer
to the question which had scarcely died upon her lips, she took

courage, and with her eyelashes sweeping her varying cheek, in a
low, yet steady voice, said:
Mr. Temple, I was asking Mr. Wynne a question, to which for some
reason he did not seem able or willing to reply; will you tell me
whether you ever knew Eugene Trevor?
An instant's pause—then, in a tone in which, though calm, there was
something unnatural and strange in the sound, there came the
laconic reply—I did.
And then there was a solemn pause. For what could Eustace Trevor
add—how reply to the mute but eager questioning of those eyes,
now fixed intently upon him, as if in the verdict of his lips there lay
more power to ease the heart of its blind fears and nameless
misgivings—more in one calm word of his
Than all the world's defied rebuke.
Therefore, though Mary held her breath, hoping, longing that he
should proceed, yet shrinking from more direct inquiry, there he
stood, with lips compressed and stern averted eyes; no marble
statue could have remained more mute; till to break the ominous
and oppressive silence, Mary pronounced the name of Eustace
Trevor.
Then, indeed, her listener's eyes relaxed their fixed expression—a
sudden glow lit up his countenance.
In a low, deep tone, and with a soft, melancholy smile, he
demanded:
And what, Miss Seaham, of Eustace Trevor?
What of him? Oh! Mr. Temple, all—everything that you may know—
may have reason to suspect or conceive concerning him!
Another pause; and then the voice of Mr. Temple, with renewed
sadness replied:

What could I tell you concerning him, but that he is a wanderer
upon the face of the earth, as you—as everybody are aware.
But why—but wherefore should this be; why forsake his country, his
home, his kindred? Now, when Louis de Burgh gave me reason to
suppose all further necessity was removed, his temporary affliction
entirely subsided, why not return?
Return! interrupted the other—return with that brand—that
stigma—which once attached to his name, must mark him in the
eyes of men—a thing of suspicion, nay, of fear for ever; return,
when that return must be to hear that curse in every blast—to be
cut off from every hope, every tie which makes life beautiful to other
men, or— he paused; for he was on the point of saying, or—bitter
alternative—brand a still worse stigma on another; on one who
however unworthy of such consideration, I must still remember as
my brother. Thus he probably would have spoken, had not he been
recalled to recollection by the strange and anxious expression
depicted on Mary's countenance, and then he added, with an effort
at self-command:
The imputation of madness is a fearful thing, Miss Seaham, to be
attached to a man's name; and Eustace Trevor, unfortunate man! is
possessed of feelings most sensitive—morbidly sensitive, perhaps.
It is—it is, Mary faltered, a fearful thing if suffered to rest there;
but surely his is not the course to accomplish the removal of the
idea. Let Eustace Trevor but return—let him at least try and
experience what a brother's kindness—what a sister's love can do, to
wipe from his remembrance the morbid memory of his past
affliction; and show to the world (if he fears its altered smiles) that
the shock his noble mind sustained was but for a moment; that he is
—
But it was enough—those words, a brother's kindness—still more, a
sister's love, had thrilled acutely upon the listener's heart.

And Mary paused, startled to behold the expression in the eyes bent
so earnestly upon her.
A sister's love! what was such love to him!
However, with another strong effort he said in a voice scarce audible
from emotion, For such a sister's love, he might indeed brave and
defy the scorn—the ignominy of the universe; but, he faltered, it
cannot be.
A silence of some minutes ensued. It was broken by Mary, who said
in an anxious trembling voice,
Mr. Temple, I have a favour to ask of you: I know you are
acquainted with much of the private history of the Trevors—I am
sure you are—I therefore entreat you will speak candidly upon the
subject, and tell me your own opinion of Eugene Trevor. To you I can
speak as I feel I can to no one else. My mind of late has been
disturbed by doubts and fears upon the subject of Eugene. I know
you can, you will speak the truth; so conceal not your real opinion
from me.
Miss Seaham, excuse me, Mr. Temple replied gravely, and with a
degree of proud coldness. I must decline to speak in any way of
Eugene Trevor. It is a long time now since we have met.
Oh, why—why, faltered Mary, with clasped hands and streaming
eyes, would you too, like the rest, by your looks, even by your
silence, make me suspect the worth, the rectitude of Eugene, and
give me the miserable idea that the affection and heart's devotion
now of years have been wasted and bestowed in vain?
It was a difficult moment for that generous, noble soul. The peculiar
situation in which he was placed almost bewildered his sense of
discernment between what was right and wrong in his position, and
darkened the way before him. How act—how speak—how meet this
critical emergency?

The struggle must have been indeed intense, which enabled him at
length to rise a conqueror over the conflicting powers which beset
his soul, to subdue all selfish promptings of inferior nature—all
selfish impulses and considerations; and speak and act as one might
have spoken and acted who had never been Mary Seaham's lover, or
Eugene Trevor's injured brother.
As a brother to a well-beloved sister—or as one of his high and holy
calling might have seized that favourable opportunity for
endeavouring to turn a perplexed and trembling suppliant on his
counsel and assistance from some dangerous path or fatal delusion,
he took up the strain, and implored her not to seek from him any
further information on a subject—concerning which he must tell her
at once, that for many reasons it was impossible for him to enter—
he could not speak of Eugene Trevor. But he implored her to think
well of those warnings so strongly pressed upon her consideration
by her anxious friends—above all, by the internal evidence of her
own pure soul—against a course of action in which the peace and
happiness of her future life might be so fatally involved.
Talk not of wasted affection, he touchingly exclaimed; affection
disinterested and blameless as yours, was never wasted—never
bestowed in vain—for some good purpose, the All Wise so willed
that you should for a time bestow it, and if He ordains that its
waters should turn back, like the rain to their springs, He wills also
that they should fill them with refreshment. Miss Seaham, it is not
for me to advise you to break off your engagement with Eugene
Trevor. I am the last person in the world—situated towards you as I
have been—he added in a low sad voice, who ought to presume so
to do; but let me speak to you, as you may remember I once before
addressed you—before it had ever entered my heart to conceive you
would stand in the position you now are in towards this Eugene
Trevor. Did I not then warn you of the world into which you were
hastening so unwarily—of its sins, its sorrows, and its snares; but
still more, of its friendships, its smiles, its Judas kisses, awaiting not
alone the eagle but the dove—the holy, harmless, and undefiled?

And now do not my gloomy words find an echo in your heart? does
not that look of care, that heavy sigh, confess that it had been
better never to have tasted of the feverish joy, the unsatisfying
delight, in exchange for the peace and tranquillity you had hitherto
enjoyed? Is not your confidence disturbed—your trust shaken in the
object on whom your affections have been set? do you not fear to
lean more heavily on that reed lest it pierce you—to grasp it firmer,
lest you crush, and prove its hollowness? Oh, Miss Seaham! is not
this in some degree the case with you? if so, do not seek to dive
further into the why or the wherefore. Let God's providence have its
way, when, it seeks to turn you from a course it is not good for you
to follow. Let faith and patience have their perfect work; seek peace
and happiness from a higher, surer source than the dubious object
on which your affections have been placed.
Mr. Temple paused, but he had no reason to suppose his earnest
appeal had been as water spilt upon the ground; for something in
Mary's face—that something, which had become of late its ruling
and habitual expression, which might have seemed to breathe forth
the Psalmist's weary longing for the wings of a dove to fly away and
be at rest—at rest, from the ever receding hopes—the sickening
doubts and apprehensions—the wearying mysteries attendant on her
position, which pressed so heavily on a nature formed rather for the
peace and calm of gentle emotions, of peaceful joys, than for its
strife of passions, its storm of woes; an expression which had
appeared to Eustace Trevor to deepen as he spoke, for not for a
moment did he dare to interpret it otherwise. Never did he surmise
—never dare even to desire—that words uttered with such
disinterested and single-minded intention, and in accents tremulous
with such unselfish emotions, could in any other way affect his
listener's heart. That in that hour of languid yearning for strength
she felt that she did not possess; for rest and peace founded on
some surer basis than that reed shaken by the wind, such as her
inauspicious love had gradually assumed the semblance, she should
be most ready to lean her weary head on the noble breast, cling to
the sheltering arm of him who thus had counselled her, and placing

her destiny in his hands, ask him to guide her future course through
the deceitful bewildering mazes of this life.
But no word, no look betrayed the secret impulse of her heart; and
in the same anxious strain Eustace Trevor proceeded:
Darkly, ambiguously, I have been compelled to speak; the subject
having been, as you can bear witness, forced in a manner upon me;
yet one step further I will take, and leave the rest in the hands of
God. This ring, drawing the signet from his finger, where for the
first time since the adventure in which it had formed a part, Mary
had again seen it; keep it, he continued, in a voice tremulous with
emotion as Mary mechanically received it in her hands, looking
wonderingly and enquiringly in his face; keep it till you see him,
Eugene Trevor again; then show it to him from me—from Edward
Temple. Tell him the circumstances under which you received it, and
ask him to clear up the mystery concerning it. If he refuses, then for
his own sake as well as your own, I conjure you to bid him farewell
for ever. If on the contrary, casting off all falsehood and deceit, he
lays all before you, then—then—may Heaven direct the rest!
An hour or two after Mary had been left alone within the marble
sala, almost as in a dream, gazing upon that mysterious and
momentous ring, the little party were proceeding northwards in the
cool of the evening, in one of the hired conveyances of the country.
Mary, her brother, and Mr. Wynne occupying the interior; Mary being
only at a later stage of the journey, confirmed in her supposition of
Mr. Temple having proceeded thus far on the outside, for since he
had parted abruptly from her he had not again appeared.
Then, however, when, to change horses, they stopped before a
road-side inn, her brother suddenly touched her arm, and directed
her attention towards the spot, where in the shadow of the door, his
features only partly distinguished in the declining evening light,

stood the tall and stately figure of Temple, apparently conversing
with Mr. Wynne who had just alighted, though his eyes were fixed
earnestly in their direction.
Look, Mary, does it not strike you now?
What, Arthur?
That likeness; there just as he stands in that uncertain light?
Mary for all reply shuddered slightly, and turned away her head. The
next moment Mr. Wynne had rejoined them, and they started again.
But by the inn-door there still stood that dark figure.
Arthur, with an exclamation of surprise, put forth his head, and
inquired why they had left Mr. Temple behind.
Because—because, Mr. Wynne replied in a peculiar tone of voice,
he has taken it into his head not to travel any further with us just
now. I shall rejoin him when I have seen you safe at Genoa, for I
cannot make up my mind to part so suddenly with my two dear
children. Temple desired me to bid you good bye, Arthur, for he has
no great fancy for leave-takings, at any time; and I was to say
farewell for him to you too, Miss Mary.
This he said in a more serious manner, taking Mary's hand as he
spoke, and gazing earnestly into her face. The hand he held was
very cold, and on the pale face there was a strange and anxious
expression; but whilst Arthur was loud in his professions of surprise
and regret at this unexpected deprivation, Mary uttered no word of
astonishment or regret.

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