Fudamentals of Mathematics

Odlomak :

Preface

The first dilemma of the authors, when preparing this textbook was its fu-ture volume. Most textbooks are thick, with plenty of material, in most casesmore than it would be presented in lectures. The reader, either lecturer or stu-dent, has to make certain choice for every particular course and to accordinglyadjust the order of its presenting. For that reason, fat books usually serve as en-cyclopedia, where from certain knowledge might be sometimes extracted. Sincethe Internet can serve the same purpose, who reads fat books these days? Forthat reason, we have chosen this book to be thin. However, the majority of thinmathematical books provide only cursory information. In most cases, terms andstatements are distorted and only short and trivial facts are being proven, moreserious omitted or, in the best case, only mentioned due the prejudice that math-ematical rigor is not for beginners and non-mathematicians, that they are notable to comprehend more complex statements or their proofs, in these books.In such a manner, mathematics is reduced to usage of prefabricated boring pat-terns. But the whole truth is that, without thorough insight of the proof details,it is often not possible to achieve neither the full comprehension nor the correctapplication of mathematics, particularly in other disciplines!

The aim of this book was to rectify these shortcomings. With such an ap-proach this textbook appeared, small in volume, but deep in its contents. It con-tains fundamental algebraic mathematical knowledge, necessary to students ofstarting years of mathematics, engineering and applied programs. The explica-tion is direct, rigorous and short. As in a hunt, the ,,dragon” is hit by the spear,and further ,,cutting” is left to an intelligent reader! The expected entering levelof knowledge is standard understanding of elementary mathematics. It is thensolidified here and broadened in a modern and scrupulous way.

This book has 182 pages. The material is divided into nine chapters: Ele-ments of mathematical logic, Set theory, Relations, Functions, Elements of com-binatorics, Algebraic structures, Field of real numbers, Field of complex num-bers and Elements of linear algebra. The recommended order of presenting thematerial is just as it is here, although we have tried, wherever possible, to makechapters independent of each other. One semester with two hours of lectures,twice a week, is necessary to cover this material in class.

The book has the following specific qualities, in difference to other text-books prepared for beginners and non-mathematicians, besides the straightfor-ward style of explication:

– all terms and statements are rigorously and correctly presented;
– many difficult statements were completely proved in the shortest, oftenoriginal, way, while easier ones are mostly left to the reader as exercises;
– aside to the answer to the standard question ,,how?”, we did our best toanswer to ,,why?”, too;
– the material is illustrated using leading, as well as non-standard, examples(different logic and combinatorial problems), which can additionally mo-tivate students, showing that the presented theory is very useful and thatmathematical knowledge can be transferred from one area to the other.In such a manner, accompanying examples and exercises that follow, nextto the clarification of introduced terms, have the role to make a readercurious and sometimes unexpectedly show applicability of the presentedknowledge;
– all important parts of the elementary mathematics are re-introduced herein original, short and strict way, because, in most cases, the previous knowl-edge base from elementary and higher education (real and complex num-bers, quadratic equations, trigonometric functions, roots,...) is relativelypoor.

We hope that, besides students, this book will prove useful to lecturers, too.It can serve as an indication how in limited number of lectures all the beautyand usefulness of mathematics could be presented, without loss of its rigor, andyet not ,,alienating” students from mathematics.

We thank our reviewers, Prof. dr S. Pilipovic, Academician and Prof. dr G.Milovanovic, Academician, as well as, Prof dr S Radjenovic, dr J. Vukmirovic,Prof. Emeritus dr C. Dolocanin, who, through numerous suggestions, substan-tially influenced the quality of this book. We extend our gratitude to Prof. dr M.Kostic, Rector of the State University of Novi Pazar who contributed to the pub-lishing of the book. We also thank Prof. dr R. Nedeljkovic who, next to translat-ing it into English, also did certain technical and substantial corrections of thismanuscript and to the lector Sibela Eminovic.

Novi Pazar,
October 2015

Authors