Zakon E. - Basic Math Conecpts.pdf

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The Zakon Series on Mathematical Analysis
Basic Concepts of Mathematics
Mathematical Analysis I
(in
preparation)
Mathematical Analysis II
(in
preparation)
9 781931 705004
The Zakon Series on Mathematical Analysis
Basic Concepts of
Mathematics
Elias Zakon
University of Windsor
The Trillia Group
West Lafayette, IN
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Basic Concepts of Mathematics
c 1973 Elias Zakon
c 2001 Bradley J. Lucier and Tamara Zakon
ISBN 1-931705-00-3
Published by The Trillia Group, West Lafayette, Indiana, USA
First published: May 26, 2001. This version released: October 29, 2001.
Technical Typist: Judy Mitchell. Copy Editor: John Spiegelman. Logo: Miriam Bogdanic.
The phrase “The Trillia Group” and The Trillia Group logo are trademarks of The Trillia
Group.
This book was prepared by Bradley J. Lucier and Tamara Zakon from a manuscript
prepared by Elias Zakon. We intend to correct and update this work as needed. If you notice
any mistakes in this work, please send e-mail to
lucier@math.purdue.edu
and they will be
corrected in a later version.
Half the proceeds from the sale of this book go to the
Elias Zakon Memorial Scholarship
fund at the University of Windsor, Canada, funding scholarships for undergraduate students
majoring in Mathematics and Statistics.
Preface
This text helps the student complete the transition from purely manipulative
to rigorous mathematics. It spells out in all detail what is often treated too
briefly or vaguely because of lack of time or space. It can be used either for sup-
plementary reading or as a half-year course. It is self-contained, though usually
the student will have had elementary calculus before starting it. Without the
“starred” sections and problems, it can be (and
was)
taught even to freshmen.
The three chapters are fairly independent and, with small adjustments, may
be taught in arbitrary order. The chapter on
n-space
“imitates” the geometry
of lines and planes in 3-space, and ensures a thorough review of the latter, for
students who may not have had it. A wealth of problems, some simple, some
challenging, follow almost every section.
Several years’ class testing led the author to these conclusions:
(1) The earlier such a course is given, the more time is gained in the follow-
up courses, be it algebra, analysis or geometry. The longer students
are taught “vague analysis”, the harder it becomes to get them used to
rigorous proofs and formulations and the harder it is for them to get rid of
the misconception that mathematics is just memorizing and manipulating
some formulas.
(2) When teaching the course to freshmen, it is advisable to start with Sec-
tions 1–7 of Chapter 2, then pass to Chapter 3, leaving Chapter 1 and
Sections 8–10 of Chapter 2 for the end. The students should be urged to
preread
the material to be taught next. (Freshmen must
learn
to read
mathematics by
rereading
what initially seems “foggy” to them.) The
teacher then may confine himself to a brief summary, and
devote most
of his time to solving as many problems
(similar
to those assigned
)
as
possible.
This is absolutely necessary.
(3) An early and constant use of logical quantifiers (even in the text) is ex-
tremely useful. Quantifiers are there to stay in mathematics.
(4) Motivations are necessary and good, provided they are brief and do not
use terms that are not yet clear to students.

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