Table Of ContentINTRODUCTION
TO MODERN
MATHEMATICS
HELENA RASIOWA
University of Warsaw
1973
N O R T H - H O L L A ND P U B L I S H I NG C O M P A NY
A M S T E R D A M · L O N D ON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
NEW YORK
Translated by
Olgierd Wojtasiewicz
Copyright 1973 by
PANSTWOWE WYDAWNICTWO NAUKOWE
(PWN — POLISH SCIENTIFIC PUBLISHERS)
Warszawa
All rights reserved
This book is a translation from the original Polish
Wstep do matematyki wspolczesnej
published by PWN — Polish Scientific Publishers, 1971
in the series "Biblioteka Matematyczna"
The English edition of this book has been published by PWN jointly with
NORTH-HOLLAND PUBLISHING COMPANY
Amsterdam
ISBN 0 7204 2067 9
Library of Congress Catalogue Card Number 12-88575
PRINTED IN POLAND
(D.R.P.)
FOREWORD
Those who begin to study mathematics usually find it difficult to
develop the habit of strictly formulating the ideas to be expressed, to
learn the methods of correct reasoning, and to comprehend the basic
concepts of mathematics. These difficulties seem to be caused, first,
by the lack of adequate training in mathematical logic, that is the
discipline whose tasks is to study deductive reasoning employed in
proving mathematical theorems; second, by the ignorance of the basic
concepts and methods used in set theory, now commonly applied in
all branches of mathematics and serving as the basis for introducing
and explaining fundamental mathematical concepts (relations, map-
pings, etc.); third, by the ignorance of the basic concepts of abstract
algebra, a discipline which has been developing vigorously and is now
affecting all the remaining branches of mathematics.
The present book covers those elements of mathematical logic,
set theory, and abstract algebra which will enable the reader to study
modern mathematics, which explains the title. The book has developed
from my lectures on Introduction to mathematics, which I have given
in Warsaw University for a couple of years, and is intended mainly
as a freshmen course in mathematics. Its scope goes much beyond
the Introduction to mathematics as formulated in the curriculum, and
this is why it may prove useful to other readers as well—those study-
ing engineering, natural science, and the humanities—who want to
prepare for advanced mathematical studies or to become familiar with
elements of mathematical logic, set theory, and the basic concepts
of abstract algebra. The present author's intention was to make the
book form a complete whole and to encourage at least some readers
to a further study of those branches of mathematics.
The exposition of the subject matter, as given in the present book,
differs from the traditional approach. First of all, set theory is not
preceded by elements of mathematical logic. The present author's
VI FOREWORD
experience has shown that a beginner finds set theory easier than
mathematical logic. Moreover, if the concepts of logic are applied to
set theory at a too early stage, the reader becomes accustomed to
a purely mechanical formulation of proofs and does not develop
mathematical intuition in grasping the concepts and theorems of set
theory. A further argument in favour of the order of presentation
of the subject matter is that the best way of demonstrating the applica-
tion of the concepts of logic in defining mathematical concepts and in
proving theorems is to do this when the student already has acquired
some knowledge of advanced mathematics. In view of the above the
present author has decided to discuss the elements of set theory before
the elements of logic. But, to make the exposition clearer and to ac-
custom the reader to logical symbolism, the author introduces that
symbolism gradually, beginning with the first sections, and applies it
systematically. The elements of mathematical logic are presented mainly
from the point of view of their applications in mathematics, chiefly
in proving theorems. This is why a formalization of logic is avoided
and more attention is paid to the rules of inference than to the laws
of logic. This applies in particular to the propositional calculus.
The book consists of 14 chapters. Chapters I and III to XI cover
the elements of set theory. In view of the elementary character of the
book an intuitive concept of function (mapping, transformation) is
first introduced in Chapter III, and a precise definition of a function
is to be found only in Chapter V. The material has been selected from
the point of view of its usefulness in the study of other branches of
mathematics. This is why the arithmetic of cardinals and ordinals has
been completely disregarded. A terminological change has been in-
troduced after Bourbaki: partially ordered sets are termed ordered
sets. Chapter II is concerned with mathematical induction and proofs
by induction, Chapter XII and XIII cover the elements of mathematical
logic. The propositional calculus, because of its elementary character,
is treated much more comprehensively than is the functional calculus.
The last section of Chapter XII illustrates a formal approach to the
propositional calculus and includes a simple proof of the completeness
theorem. The functional calculus is not presented as a formal system,
and its exposition is confined to those laws and rules of inference which
FOREWORD vii
are most frequently used in mathematical reasoning, and to many
examples that point to its applications in mathematics. Chapter XIV
is essentially only a supplement to the book and explains the basic
concepts of abstract algebra, such as subalgebra, homomorphism, iso-
morphism, congruence, etc.
When working on this book the author has made use of her lecture
notes Introduction to Mathematical Logic and Set Theory (in Polish),
covering her lectures at a Course in Applications of Mathematics,
organized in 1965 by the Polish Academy of Sciences, Institute of
Mathematics. These lecture notes have since been published by the
Polish Academy of Sciences Division for Training Research Staff.
In writing the part of the book concerned with set theory the author
has drawn largely from Professor K. Kuratowski's Introduction to
Set Theory and Topology and from Set Theory by Professors A. Mos-
towski and K. Kuratowski (both now available in English-language
versions).
The author is indebted to Professors S. Hartman and A. Mostowski
for their suggestions about the manuscript version of the present book;
their criticism has helped her to improve the original text and to remove
its various shortcomings.
Thanks are also due to Professor M. Stark for his suggestion that
the present book be written and published.
HELENA RASIOWA
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CHAPTER I
THE ALGEBRA OF SETS
1. The concept of set
The concept of set is one of the fundamental concepts of mathematics.
As examples of sets one may quote: the set of all books in a given
library; the set of all letters of the Greek alphabet; the set of all integers;
the set of all sides of a given polygon; the set of all circles on a given
plane.
The branch of mathematics concerned with the study of the general
properties of sets, regardless of the nature of the objects which form
those sets, is termed set theory1) and is regarded as the foundation
of modern mathematics. This discipline was founded by Georg Cantor
in the years 1871-1883.
The objects which belong to a given set are called its elements.
The statement that an element a belongs to a set A (or: that a is an
element of a set A) is written:
(1) aeA2),
while a e A, b e A, c e A will often be abbreviated into: a, b, c e A.
The statement that a does not belong to a set A (i.e., a is not an
element of the set A) will be written
(2) αφ A or ~(aeA).
The symbol ~ will always stand for not or it is not the case that 3)
It is convenient to introduce in mathematics the concept of empty
set, i.e., the set which has no elements. It may be said, for instance,
1) Special reference should be made to the role of Polish mathematicians in de-
veloping this discipline, especially to the numerous papers by W. Sierpinski.
2) The symbol e was introduced by G. Peano; it is the first letter of the Greek
word εστί (is).
3) The symbol ~ is a distorted letter Ν (from nego, Lat. I deny).
2 I. ALGEBRA OF SETS
that "the set of all real roots of the equation x 2 +1 =0 is empty"
instead of "there does not exist any real number which is a root of
the equation x2 +1 = 0".
The empty set is denoted by O.
The set whose all elements are a , ..., a will be denoted by
1 n
(3) {a...,a}.
l9 n
A set may also consist of one element. For instance, the set of all
even prime numbers has exactly one element, namely the number 2.
A set whose only element is a will, by analogy to (3), be denoted by
(4) {a}.
If every element of a set A is an element of a set B, then we say that
A is a subset of Β We also say that the set A is contained in the set Β
or that Β contains A, which is written
A c Β or Β ^ A.
Fig. 1
The symbol c is called the symbol of inclusion. By definition, A c Β
if and only if the following condition is satisfied for every x: if χ e A,
then xeB2). Hereafter, the words if... then... will often be replaced
by the symbol =>, and the words if and only if by the symbol o.
Accordingly the above formulation may be written in symbols as:
(5) (A a B)o (for every χ: χ e A => χ e B).
1) If at the same time the sets A, Β are not identical (cf. p. 3), it is said that A
is a proper subset of B.
2) The phrase: α if and only if β (where α and β are any formulas that have the
form of statements) means : if a, then β, and if β, then a.
1. CONCEPT OF SET 3
Examples
The set of all integers is contained in the set of all rational numbers,
since every integer is a rational number. The set A = {1, 2} is contained
in the set Β = {1,2,3}, since 1 eB and 2eB. Let C and D be, respec-
tively, areas of the circles shown in Fig. 1. The set C is a subset of the
set D since every element of the set C is an element of the set D. The
9
set of all irrational numbers is contained in the set of all real numbers,
since every irrational number is a real number.
The statement that A is not a subset of Β is written
Α φ Β or Β φ A.
The following notations are also used:
~(A cz B) or ~(B ZD A).
It follows from the definition of a subset that Α φ Β if and only
if not every element of the set A is an element of the set B that is, there
9
exists in the set A an element which is not an element of B. In symbolic
notation:
(6) ~(A ci B)o (there is an χ such that: χ e A and e B)).
Examples
Figs. 2-4 shows examples of sets A and Β such that ~(A c B).
The sets A and Β are represented by areas of circles.
Fig. 2 Fig. 3 Fig. 4
The set of all integers divisible by 3 is not contained in the set of
all integers divisible by 6, since there exists an integer divisible by 3
which is not divisible by 6, e.g., the number 9. The number 9 belongs
to the former set, but not to the latter.
The sets A and Β are identical if and only if they have the same ele-
ments. This is written as follows:
(7) {A = B) <=> (for every χ: χ e Α ο χ e B).
4 I. ALGEBRA OF SETS
Example
Let A be the set of all integers that are divisible by both 2 and 3,
and let Β be the set of all integers that are divisible by 6. The sets A
and Β are identical, since an integer is divisible by both 2 and 3 if and
only if it is divisible by 6, and hence the sets A and Β have the same
elements.
It follows from the definition of a subset that
1.1. For any sets A, B, C:
(8) OŒA,
(9) A cz A,
(10) if A cz Β and Β cz C, then A cz C,
(11) if A cz Β and Β cz A, then A = B
y
(12) if Αφ B, then Α φ Β or Β φ A,
Formula (8) states that the empty set is contained in every set. Since
the empty set does not have any elements, the condition that every
element of the set Ο is an element of a set A is satisfied
Formula (9) states that every set is a subset of itself. In fact, every
element of a set A is an element of A.
Formula (10) is the law of transitivity for the relation of inclusiont
To prove it let us assume that A cz Β and Β cz C. Then every elemen.
of the set A is an element of the set B, and every element of the set Β
is an element of the set C. Hence it follows that every element of the
set A is an element of the set C, and hence A cz C.
To prove (11) let us assume that A cz Β and Β cz A. Hence every
element of the set A is an element of the set B, and every element of
the set Β is an element of the set A. Thus the sets A and Β have the
same elements, that is, they are identical.
Formula (12) follows from formula (11). Should it be that A cz Β
and Β cz A, then by (11) the sets A and Β would be identical, contrary
to the assumption that Α φ Β.
Formula (11) is often used in proving the identity of sets.
l) Cf. example 1 in Chapter XIII, Section 1, p. 236.
