Mathematics of economics and business

Mathematics of Economics and
Business

Knowledge of mathematical methods has become a prerequisite for all students who wish to
understand current economic and business literature. This book covers all the major topics
required to gain a firm grounding in the subject, such as sequences, series, application in
finance, functions, differentiations, differential and difference equations, optimizations with
and without constraints, integrations and much more.
Written in an easy and accessible style with precise definitions and theorems,
Mathematics of Economics and Business contains exercises and worked examples, as well
as economic applications. This book will provide the reader with a comprehensive
understanding of the mathematical models and tools used in both economics and
business.
Frank Werner is Extraordinary Professor of Mathematics at Otto-von-Guericke University
in Magdeburg, Germany.
Yuri N. Sotskov is Professor at the United Institute of Informatics Problems, National
Academy of Science of Belarus, Minsk.



Mathematics of Economics
and Business

Frank Werner and Yuri N. Sotskov


First published 2006
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
Simultaneously published in the USA and Canada

by Routledge
270 Madison Ave, New York, NY10016
Routledge is an imprint of the Taylor & Francis Group
© 2006 Frank Werner and Yuri N. Sotskov
This edition published in the Taylor & Francis e-Library, 2006.
“To purchase your own copy of this or any of Taylor & Francis or Routledge’s
collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”
All rights reserved. No part of this book may be reprinted or reproduced
or utilised in any form or by any electronic, mechanical, or other means,
now known or hereafter invented, including photocopying and recording,
or in any information storage or retrieval system, without permission in
writing from the publishers.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
A catalog record for this title has been requested
ISBN10: 0–415–33280–X (hbk)
ISBN10: 0–415–33281–8 (pbk)
ISBN13: 9–78–0–415–33280–4 (hbk)
ISBN13: 9–78–0–415–33281–1 (pbk)


Contents

Preface
List of abbreviations
List of notations
1 Introduction
1.1

1.2

1.3
1.4

2.2

2.3

1

Logic and propositional calculus 1
1.1.1 Propositions and their composition 1
1.1.2 Universal and existential propositions 7
1.1.3 Types of mathematical proof 9
Sets and operations on sets 15
1.2.1 Basic definitions 15
1.2.2 Operations on sets 16
Combinatorics 26
Real numbers and complex numbers 32
1.4.1 Real numbers 32
1.4.2 Complex numbers 47

2 Sequences; series; finance
2.1

ix
xiii
xv

Sequences 61
2.1.1 Basic definitions 61
2.1.2 Limit of a sequence 65
Series 71
2.2.1 Partial sums 71
2.2.2 Series and convergence of series 73
Finance 80
2.3.1 Simple interest and compound interest 80
2.3.2 Periodic payments 85
2.3.3 Loan repayments, redemption tables 90
2.3.4 Investment projects 97
2.3.5 Depreciation 101

61


vi Contents
3 Relations; mappings; functions of a real variable
3.1
3.2
3.3

Relations 107
Mappings 110
Functions of a real variable 116
3.3.1 Basic notions 117
3.3.2 Properties of functions 121
3.3.3 Elementary types of functions 126

4 Differentiation

4.1

4.2
4.3
4.4
4.5

4.6
4.7
4.8

5.3
5.4
5.5

5.6

197

Indefinite integrals 197
Integration formulas and methods 198
5.2.1 Basic indefinite integrals and rules 198
5.2.2 Integration by substitution 200
5.2.3 Integration by parts 204
The definite integral 209
Approximation of definite integrals 215
Improper integrals 219
5.5.1 Infinite limits of integration 219
5.5.2 Unbounded integrands 220
Some applications of integration 222

5.6.1 Present value of a continuous future income flow 222
5.6.2 Lorenz curves 224
5.6.3 Consumer and producer surplus 225

6 Vectors
6.1
6.2

148

Limit and continuity 148
4.1.1 Limit of a function 148
4.1.2 Continuity of a function 151
Difference quotient and the derivative 155
Derivatives of elementary functions; differentiation rules 158
Differential; rate of change and elasticity 164
Graphing functions 168
4.5.1 Monotonicity 168
4.5.2 Extreme points 169
4.5.3 Convexity and concavity 175
4.5.4 Limits 178
4.5.5 Further examples 181
Mean-value theorem 184
Taylor polynomials 186
Approximate determination of zeroes 189

5 Integration
5.1
5.2

107

Preliminaries 230
Operations on vectors 233

230


Contents vii
6.3
6.4

Linear dependence and independence 240
Vector spaces 244

7 Matrices and determinants
7.1
7.2
7.3
7.4
7.5
7.6

Matrices 253
Matrix operations 258
Determinants 263
Linear mappings 271
The inverse matrix 273
An economic application: input–output model 277

8 Linear equations and inequalities
8.1

8.2

368

Eigenvalues and eigenvectors 368
Quadratic forms and their sign 376

11 Functions of several variables
11.1
11.2
11.3
11.4
11.5
11.6

328

Preliminaries 328
Graphical solution 330
Properties of a linear programming problem; standard form 334
Simplex algorithm 339
Two-phase simplex algorithm 350
Duality; complementary slackness 357
Dual simplex algorithm 363

10 Eigenvalue problems and quadratic forms
10.1

10.2

287

Systems of linear equations 287
8.1.1 Preliminaries 287
8.1.2 Existence and uniqueness of a solution 290
8.1.3 Elementary transformation; solution procedures 292
8.1.4 General solution 302
8.1.5 Matrix inversion 306
Systems of linear inequalities 308
8.2.1 Preliminaries 308
8.2.2 Properties of feasible solutions 309
8.2.3 A solution procedure 315

9 Linear programming
9.1
9.2
9.3
9.4
9.5
9.6
9.7

253

Preliminaries 383
Partial derivatives; gradient 387
Total differential 394
Generalized chain rule; directional derivatives 397

Partial rate of change and elasticity; homogeneous functions 402
Implicit functions 405

383


viii Contents
11.7

11.8

11.9

Unconstrained optimization 409
11.7.1 Optimality conditions 409
11.7.2 Method of least squares 419
11.7.3 Extreme points of implicit functions 423
Constrained optimization 424
11.8.1 Local optimality conditions 424
11.8.2 Global optimality conditions 434
Double integrals 436

12 Differential equations and difference equations

444

12.1

Differential equations of the first order 445
12.1.1 Graphical solution 445

12.1.2 Separable differential equations 447
12.2 Linear differential equations of order n 451
12.2.1 Properties of solutions 451
12.2.2 Differential equations with constant coefficients 454
12.3 Systems of linear differential equations of the first order 461
12.4 Linear difference equations 472
12.4.1 Definitions and properties of solutions 472
12.4.2 Linear difference equations of the first order 474
12.4.3 Linear difference equations of the second order 478
Selected solutions
Literature
Index

486
511
513


Preface

Today, a firm understanding of mathematics is essential for any serious student of economics.
Students of economics need nowadays several important mathematical tools. These include
calculus for functions of one or several variables as well as a basic understanding of
optimization with and without constraints, e.g. linear programming plays an important
role in optimizing production programs. Linear algebra is used in economic theory and
econometrics. Students in other areas of economics can benefit for instance from some
knowledge about differential and difference equations or mathematical problems arising in
finance. The more complex economics becomes, the more deep mathematics is required and
used. Today economists consider mathematics as the most important tool of economics and
business. This book is not a book on mathematical economics, but a book on higher-level

mathematics for economists.
Experience shows that students who enter a university and specialize in economics vary
enormously in the range of their mathematical skills and aptitudes. Since mathematics is
often a requirement for specialist studies in economics, we felt a need to provide as much
elementary material as possible in order to give those students with weaker mathematical
backgrounds the chance to get started. Using this book may depend on the skills of readers and
their purposes. The book starts with very basic mathematical principles. Therefore, we have
included some material that covers several topics of mathematics in school (e.g. fractions,
powers, roots and logarithms in Chapter 1 or functions of a real variable in Chapter 3). So
the reader can judge whether or not he (she) is sufficiently familiar with mathematics to be
able to skip some of the sections or chapters.
Studying mathematics is very difficult for most students of economics and business.
However, nowadays it is indeed necessary to know a lot of results of higher mathematics
to understand the current economic literature and to use modern economic tools in practical
economics and business. With this in mind, we wrote the book as simply as possible. On the
other hand, we have presented the mathematical results strongly correct and complete, as is
necessary in mathematics. The material is appropriately ordered according to mathematical
requirements (while courses, e.g. in macroeconomics, often start with advanced topics such
as constrained optimization for functions of several variables). On the one hand, previous
results are used by later results in the text. On the other hand, current results in a chapter
make it clear why previous results were included in the book.
The book is written for non-mathematicians (or rather, for those people who only want
to use mathematical tools in their practice). It intends to support students in learning the
basic mathematical methods that have become indispensable for a proper understanding
of the current economic literature. Therefore, the book contains a lot of worked examples
and economic applications. It also contains many illustrations and figures to simplify the


x Preface
mathematical techniques used and show how mathematical results may be used in economics

and business. Some of these examples have been taken from former examinations (at the
Otto-von-Guericke University of Magdeburg), and many of the exercises given at the end of
each chapter have been used in the tutorials for a long time. In this book, we do not show how
the mathematical results have been obtained and proved, but we show how they may be used
in real-life economics and business. Therefore, proofs of theorems have been skipped (with
a few exceptions) so that the volume of the book does not substantially exceed 500 pages,
but in spite of the relatively short length the book includes the main mathematical subjects
useful for practical economics and an efficient business.
The book should serve not only as a textbook for a course on mathematical methods for
students, but also as a reference book for business people who need to use higher-level
mathematics to increase profits. (Of course, one will not increase profit by solving e.g.
a differential equation, but one can understand why somebody has increased profits after
modelling a real process and finding a solution for it.) One of the purposes of this book is to
introduce the reader to the most important mathematical methods used in current economic
literature. We also provide an introduction to the close relationship between mathematical
methods and problems arising in the economy. However, we have included only such
economic applications as do not require an advanced knowledge of economic disciplines,
since mathematics is usually taught in the first year of studies at university.
The reader needs only knowledge of elementary mathematics from secondary school to
understand and use the results of the book, i.e. the content is self-sufficient for understanding.
For a deeper understanding of higher mathematics used in economics, we also suggest a small
selection of German and English textbooks and lecture materials listed in the literature section
at the end of the book. Some of these books have been written at a comparable mathematical
level (e.g. Opitz, Mathematik; Simon and Blume, Mathematics for Economists; Sydsaeter
and Hammond, Mathematics for Economic Analysis) while others are more elementary in
style (e.g. Misrahi and Sullivan, Mathematics and Finite Mathematics; Ohse, Mathematik
für Wirtschaftswissenschaftler; Rosser, Basic Mathematics for Economists). The booklets
(Schulz, Mathematik für wirtschaftswissenchaftliche Studiengänge; Werner, Mathematics
for Students of Economics and Management) contain most important definitions, theorems
of a one-year lecture course in mathematics for economists in a compact form and

they sketch some basic algorithms taught in the mathematics classes for economists at
the Otto-von-Guericke University of Magdeburg during recent decades. Bronstein and
Semandjajew, Taschenbuch der Mathematik, and Eichholz and Vilkner, Taschenbuch der
Wirtschaftsmathematik, are well-known handbooks of mathematics for students. Varian,
Intermediate Microeconomics, is a standard textbook of intermediate microeconomics, where
various economic applications of mathematics can be found.
The book is based on a two-semester course with four hours of lectures per week which
the first author has given at the Otto-von-Guericke University of Magdeburg within the last
ten years. The authors are indebted to many people in the writing of the book. First of
all, the authors would like to thank Dr Iris Paasche, who was responsible for the tutorials
from the beginning of this course in Magdeburg. She contributed many suggestions for
including exercises and for improvements of the contents and, last but not least, she prepared
the answers to the exercises. Moreover, the authors are grateful to Dr Günther Schulz for
his support and valuable suggestions which were based on his wealth of experience in
teaching students of economics and management at the Otto-von-Guericke University of
Magdeburg for more than twenty years. The authors are grateful to both colleagues for their
contributions.


Preface xi
The authors also thank Ms Natalja Leshchenko of the United Institute of Informatics
Problems of the National Academy of Sciences of Belarus for reading the whole manuscript
(and carefully checking the examples) and Mr Georgij Andreev of the same institute for
preparing a substantial number of the figures. Moreover, many students of the International
Study Programme of Economics and Management at the Otto-von-Guericke University of
Magdeburg have read single chapters and contributed useful suggestions, particularly the
students from the course starting in October 2002. In addition, the authors would like to
express their gratitude to the former Ph.D. students Dr Nadezhda Sotskova and Dr Volker
Lauff, who carefully prepared in the early stages a part of the book formerly used as printed
manuscript in LATEX and who made a lot of constructive suggestions for improvements.

Although both authors have taught in English at universities for many years and during
that time have published more than 100 research papers in English, we are nevertheless
not native speakers. So we apologize for all the linguistic weaknesses (and hope there are
not too many). Of course, for all remaining mathematical and stylistic mistakes we take
full responsibility, and we will be grateful for any further comments and suggestions for
improvements by readers for inclusion in future editions (e-mail address for correspondence:
). Furthermore, we are grateful to Routledge
for their pleasant cooperation during the preparation of the book. The authors wish all readers
success in studying mathematics.
We dedicate the book to our parents Hannelore Werner, Willi Werner, Maja N. Sotskova
and Nazar F. Sotskov.
F.W.
Y.N.S.



Abbreviations

p.a.
NPV
resp.
rad
l
m
cm
km
s
EUR
LPP
s.t.

bv
nbv

per annum
net present value
respectively
radian
litre
metre
centimetre
kilometre
second
euro
linear programming problem
subject to
basic variables of a system of linear equations
non-basic variables of a system of linear equations



Notations

A
A∧B
A∨B
A =⇒ B
A ⇐⇒ B
A(x)

negation of proposition A

conjunction of propositions A and B
disjunction of propositions A and B
implication (if A then B)
equivalence of propositions A and B
universal proposition

x

A(x)

existential proposition

x

a∈A
b∈
/A
∅
|A|

P(A)
A⊆B
A∪B
A∩B
A\B
A×B
n

Ai

a is an element of set A
b is not an element of set A
empty set
cardinality of a set A (if A is a finite set, then |A| is equal to the
number of elements in set A), the same notation is used for the
determinant of a square matrix A
power set of set A
set A is a subset of set B
union of sets A and B
intersection of sets A and B
difference of sets A and B
Cartesian product of sets A and B
Cartesian product of sets A1 , A2 , . . . , An

i=1

An

n

Cartesian product

Ai , where A1 = A2 = . . . = An = A

i=1

n!
n
k
n = 1, 2, . . . , k

N
N0
Z

n factorial: n! = 1 · 2 · . . . · (n − 1) · n
binomial coefficient:
n
n!
=
k
k! · (n − k)!
equalities n = 1, n = 2, . . . , n = k
set of all natural numbers: N = {1, 2, 3, . . .}
union of set N with number zero: N0 = N ∪ {0}
union of set N0 with the set of all negative integers


xvi List of notations
Q
R
R+
(a, b)
[a, b]
±
∓
|a|
≈
=
π
e

∞
√
a
exp
log
lg
ln

set of all rational numbers, i.e. set of all fractions p/q with p ∈ Z
and q ∈ N
set of all real numbers
set of all non-negative real numbers
open interval between a and b
closed interval between a and b
denotes two cases of a mathematical term: the first one with sign +
and the second one with sign −
denotes two cases of a mathematical term: the first one with sign −
and the second one with sign +
absolute value of number a ∈ R √
sign of approximate equality, e.g. 2 ≈ 1.41
sign ‘not equal’
irrational number equal to the circle length divided by the diameter
length: π ≈ 3.14159...
Euler’s number: e ≈ 2.71828...
infinity
square root of a
notation used for the exponential function with base e: y = exp(x) = ex
notation used for the logarithm: if y = loga x, then ay = x
notation used for the logarithm with base 10: lg x = log10 x
notation used for the logarithm with base e: ln a = loge x

summation sign:
n

a i = a1 + a 2 + · · · + a n
i=1

product sign:
n

a i = a1 · a 2 · . . . · a n
i=1

i
C
|z|
{an }
{sn }
lim
aRb
aRb
R−1
S ◦R
f :A→B
b = f (a)
f −1
g◦f
Df
Rf
y = f (x)

imaginary unit: i2 = −1
set of all complex numbers: z = a + bi, where a and b are real numbers
modulus of number z ∈ C
sequence: {an } = a1 , a2 , a3 , . . . , an , . . .
series, i.e. the sequence of partial sums of a sequence {an }
limit sign
a is related to b by the binary relation R
a is not related to b by the binary relation R
inverse relation of R
composite relation of R and S
mapping or function f ∈ A × B: f is a binary relation
which assigns to a ∈ A exactly one b ∈ B
b is the image of a assigned by mapping f
inverse mapping or function of f
composite mapping or function of f and g
domain of a function f of n ≥ 1 real variables
range of a function f of n ≥ 1 real variables
y ∈ R is the function value of x ∈ R, i.e. the value of
function f at point x


List of notations xvii
deg P
x → x0
x → x0 + 0
x → x0 − 0
f
f (x), y (x)
f (x), y (x)
dy, df

≡

degree of polynomial P
x tends to x0
x tends to x0 from the right side
x tends to x0 from the left side
derivative of function f
derivative of function f with y = f (x) at point x
second derivative of function f with y = f (x) at point x
differential of function f with y = f (x)
sign of identical equality, e.g. f (x) ≡ 0 means that equality
f (x) = 0 holds for any value x
ρf (x0 )
proportional rate of change of function f at point x0
elasticity of function f at point x0
εf (x0 )
integral sign
Rn
n-dimensional Euclidean space, i.e. set of all real n-tuples
set of all non-negative real n-tuples
Rn+
a
vector: ordered n-tuple of real numbers a1 , a2 , . . . , an
corresponding to a matrix with one column
aT
transposed vector of vector a
|a|
Euclidean length or norm of vector a
|a − b|
Euclidean distance between vectors a ∈ Rn and b ∈ Rn

a⊥b
means that vectors a and b are orthogonal
dim V
dimension of the vector space V
Am,n
matrix of order (dimension) m × n
AT
transpose of matrix A
An
nth power of a square matrix A
det A, (or |A|)
determinant of a matrix A
A−1
inverse matrix of matrix A
adj (A)
adjoint of matrix A
r(A)
rank of matrix A
x1 , x2 , . . . , xn ≥ 0 denotes the inequalities x1 ≥ 0, x2 ≥ 0, . . . , xn ≥ 0
Ri ∈ {≤, =, ≥}
means that one of these signs hold in the ith constraint
of a system of linear inequalities
z → min!
indicates that the value of function z should become minimal
for the desired solution
z → max!
indicates that the value of function z should become maximal
for the desired solution
fx (x0 , y0 )
partial derivative of function f with z = f (x, y) with

respect to x at point (x0 , y0 )
0
fxi (x )
partial derivative of function f with z = f (x1 , x2 , . . . , xn )
with respect to xi at point x0 = (x10 , x20 , . . . , xn0 )
grad f (x0 )
gradient of function f at point x0
ρf ,xi (x0 )
partial rate of change of function f with respect to xi at point x0
0
εf ,xi (x )
partial elasticity of function f with respect to xi at point x0
0
Hf (x )
Hessian matrix of function f at point x0
Q.E.D. (quod erat demonstrandum
– ‘that which was to be demonstrated’)



1

Introduction

In this chapter, an overview on some basic topics in mathematics is given. We summarize
elements of logic and basic properties of sets and operations with sets. Some comments on
basic combinatorial problems are given. We also include the main important facts concerning
number systems and summarize rules for operations with numbers.

1.1 LOGIC AND PROPOSITIONAL CALCULUS

This section deals with basic elements of mathematical logic. In addition to propositions and
logical operations, we discuss types of mathematical proofs.

1.1.1 Propositions and their composition
Let us consider the following four statements A, B, C and D.
A Number 126 is divisible by number 3.
B Equality 5 · 11 = 65 holds.
C Number 11 is a prime number.
D On 1 July 1000 it was raining in Magdeburg.
Obviously, the statements A and C are true. Statement B is false since 5 · 11 = 55. Statement
D is either true or false but today probably nobody knows. For each of the above statements
we have only two possibilities concerning their truth values (to be true or to be false). This
leads to the notion of a proposition introduced in the following definition.

Definition 1.1

A statement which is either true or false is called a proposition.

Remark For a proposition, there are no other truth values than ‘true’ (T) or ‘false’ (F)
allowed. Furthermore, a proposition cannot have both truth values ‘true’ and ‘false’ ( principle
of excluded contradiction).
Next, we consider logical operations. We introduce the negation of a proposition and connect different propositions. Furthermore, the truth value of such compound propositions is
investigated.


2 Introduction
Definition 1.2 A proposition A (read: not A) is called the negation of proposition A.
Proposition A is true if A is false. Proposition A is false if A is true.

One can illustrate Definition 1.2 by a so-called truth table. According to Definition 1.2, the

truth table of the negation of proposition A is as follows:
A

T

F

A

F

T

Considering the negations of the propositions A, B and C, we obtain:
A Number 126 is not divisible by the number 3.
B Equality 5 · 11 = 65 does not hold.
C The number 11 is not a prime number.
Propositions A and C are false and B is true.

Definition 1.3 The proposition A ∧ B (read: A and B) is called a conjunction.
Proposition A ∧ B is true only if propositions A and B are both true. Otherwise, A ∧ B
is false.

According to Definition 1.3, the truth table of the conjunction A ∧ B is as follows:
A
B
A∧B

T
T

T

T
F
F

F
T
F

F
F
F

Definition 1.4 The proposition A∨B (read: A or B) is called a disjunction. Proposition
A ∨ B is false only if propositions A and B are both false. Otherwise, A ∨ B is true.

According to Definition 1.4, the truth table of the disjunction A ∨ B is as follows:
A
B
A∨B

T
T
T

T
F
T

F
T
T

F
F
F

The symbol ∨ stands for the ‘inclusive or’ which allows that both propositions are true (in
contrast to the ‘exclusive or’, where the latter is not possible).


Introduction 3
Example 1.1 Consider the propositions M and P.
M
P

In 2003 Magdeburg was the largest city in Germany.
In 2003 Paris was the capital of France.

Although proposition P is true, the conjunction M ∧ P is false, since Magdeburg was not
the largest city in Germany in 2003 (i.e. proposition M is false). However, the disjunction
M ∨ P is true, since (at least) one of the two propositions is true (namely proposition P).

Definition 1.5 The proposition A =⇒ B (read: if A then B) is called an implication.
Only if A is true and B is false, is the proposition A =⇒ B defined to be false. In all
remaining cases, the proposition A =⇒ B is true.

According to Definition 1.5, the truth table of the implication A =⇒ B is as follows:
A

B
A =⇒ B

T
T
T

T
F
F

F
T
T

F
F
T

For the implication A =⇒ B, proposition A is called the hypothesis and proposition B is
called the conclusion. An implication is also known as a conditional statement. Next, we
give an illustration of the implication. A student says: If the price of the book is at most
20 EUR, I will buy it. This is an implication A =⇒ B with
A
B

The price of the book is at most 20 EUR.
The student will buy the book.

In the first case of the four possibilities in the above truth table (second column), the student

confirms the validity of the implication A =⇒ B (due to the low price of no more than
20 EUR, the student will buy the book). In the second case (third column), the implication is
false since the price of the book is low but the student will not buy the book. The truth value
of an implication is also true if A is false but B is true (fourth column). In our example, this
means that it is possible that the student will also buy the book in the case of an unexpectedly
high price of more than 20 EUR. (This does not contradict the fact that the student certainly
will buy the book for a price lower than or equal to 20 EUR.) In the fourth case (fifth column
of the truth table), the high price is the reason that the student will not buy the book. So in
all four cases, the definition of the truth value corresponds with our intuition.

Example 1.2 Consider the propositions A and B defined as follows:
A
B

The natural number n is divisible by 6.
The natural number n is divisible by 3.

We investigate the implication A =⇒ B. Since each of the propositions A and B can be true
and false, we have to consider four possibilities.


4 Introduction
If n is a multiple of 6 (i.e. n ∈ {6, 12, 18, . . .}), then both A and B are true. According
to Definition 1.5, the implication A =⇒ B is true. If n is a multiple of 3 but not of 6
(i.e. n ∈ {3, 9, 15, . . .}), then A is false but B is true. Therefore, implication A =⇒ B is true.
If n is not a multiple of 3 (i.e. n ∈ {1, 2, 4, 5, 7, 8, 10, . . .}), then both A and B are false, and
by Definition 1.5, implication A =⇒ B is true. It is worth noting that the case where A is true
but B is false cannot occur, since no natural number which is divisible by 6 is not divisible
by 3.

Remark
(1)
(2)
(3)
(4)
(5)
(6)

For an implication A =⇒ B, one can also say:

A implies B;
from A it follows B;
A is sufficient for B;
B is necessary for A;
A is true only if B is true;
if A is true, then B is true.

The latter four formulations are used in connection with the presentation of mathematical
theorems and their proof.

Example 1.3 Consider the propositions
H
E

Claudia is happy today.
Claudia does not have an examination today.

Then the implication H =⇒ E means: If Claudia is happy today, she does not have an
examination today. Therefore, a necessary condition for Claudia to be happy today is that
she does not have an examination today.

In the case of the opposite implication E =⇒ H , a sufficient condition for Claudia to be
happy today is that she does not have an examination today.
If both implications H =⇒ E and E =⇒ H are true, it means that Claudia is happy today if
and only if she does not have an examination today.

Definition 1.6 The proposition A ⇐⇒ B (read: A is equivalent to B) is called
equivalence. Proposition A ⇐⇒ B is true if both propositions A and B are true or
propositions A and B are both false. Otherwise, proposition A ⇐⇒ B is false.

According to Definition 1.6, the truth table of the equivalence A ⇐⇒ B is as follows:
A
B
A ⇐⇒ B

T
T
T

T
F
F

F
T
F

F
F
T

Introduction 5
Remark

For an equivalence A ⇐⇒ B, one can also say

(1) A holds if and only if B holds;
(2) A is necessary and sufficient for B.
For a compound proposition consisting of more than two propositions, there is a hierarchy
of the logical operations as follows. The negation of a proposition has the highest priority,
then both the conjunction and the disjunction have the second highest priority and finally the
implication and the equivalence have the lowest priority. Thus, the proposition
A ∧ B ⇐⇒ C
may also be written as
(A) ∧ B ⇐⇒ C.
By means of a truth table we can investigate the truth value of arbitrary compound
propositions.

Example 1.4 We investigate the truth value of the compound proposition
A ∨ B =⇒ B =⇒ A .
One has to consider four possible combinations of the truth values of A and B (each of the
propositions A and B can be either true or false):
A
B
A∨B
A∨B
A
B
B =⇒ A
A ∨ B =⇒ B =⇒ A

T
T
T
F
F
F
T
T

T
F
T
F
F
T
F
T

F
T
T
F
T
F
T
T

F
F

F
T
T
T
T
T

In Example 1.4, the implication is always true, independently of the truth values of the
individual propositions. This leads to the following definition.

Definition 1.7 A compound proposition which is true independently of the truth
values of the individual propositions is called a tautology. A compound proposition
being always false is called a contradiction.


6 Introduction
Example 1.5 We investigate whether the implication
A =⇒ B =⇒ A =⇒ B

(1.1)

is a tautology. As in the previous example, we have to consider four combinations of truth
values of propositions A and B. This yields the following truth table:
A
B
A =⇒ B
A =⇒ B
A
B
A =⇒ B

A =⇒ B =⇒ (A =⇒ B)

T
T
T
F
F
F
T
T

T
F
F
T
F
T
T
T

F
T
T
F
T
F
F
T

F

F
T
F
T
T
T
T

Independently of the truth values of A and B, the truth value of the implication considered is
true. Therefore, implication (1.1) is a tautology.

Some further tautologies are presented in the following theorem.
THEOREM 1.1

The following propositions are tautologies:

(1) A ∧ B ⇐⇒ B ∧ A,
A ∨ B ⇐⇒ B ∨ A
(commutative laws of conjunction and disjunction);
(2) (A ∧ B) ∧ C ⇐⇒ A ∧ (B ∧ C),
(A ∨ B) ∨ C ⇐⇒ A ∨ (B ∨ C)
(associative laws of conjunction and disjunction);
(3) (A ∧ B) ∨ C ⇐⇒ (A ∨ C) ∧ (B ∨ C),
(A ∨ B) ∧ C ⇐⇒ (A ∧ C) ∨ (B ∧ C)
(distributive laws).
Remark The negation of a disjunction of two propositions is a conjunction and, analogously, the negation of a conjunction of two propositions is a disjunction. We get
A ∨ B ⇐⇒ A ∧ B

and

A ∧ B ⇐⇒ A ∨ B

(de Morgan’s laws).

PROOF De Morgan’s laws can be proved by using truth tables. Let us prove the first equivalence. Since each of the propositions A and B has two possible truth values T and F, we have
to consider four combinations of the truth values of A and B:
A
B
A∨B
A∨B
A
B
A∧B

T
T
T
F
F
F
F

T
F
T
F
F
T
F

F
T
T
F
T
F
F

F
F
F
T
T
T
T