About the bookThis book tells how one may formally describe properties of the well-known relations mention­ed in the title.This example is used to clarify the transition from familiar, but imprecise concepts to strict mathematical definitions.The need for strict descriptions of the simplest relations arises in mathematical logic, cyberne­ tics, mathematical linguistics, etc. The last chapter of the book is devoted to the simplest examples from mathematical linguistics.This book was written as a popular introduction to the theory of binary relations. The binary relations studied previously from the point of view of mathematical logic’s special needs turned out to be a very simple and convenient apparatus for quite a variety of problems. The language of binary (and more general relations) is very convenient and natural for mathematical linguistics, mathematical biology*and a great many other applied (for mathematics) fields. This is very easy to explain if we say that the geometric aspect of the theory of binary relations is simply the theory of graphs. But if geometric graph theory is well-known and widely represented in the most varied kinds of literature — from popular to monographic, the algebraic aspects of the theory of relations have received almost no systematic treatment.But in spite of this, the algebra of relations can be pre­sented so comprehensibly that it could be grasped by high school students attending mathematical study circles, linguists dealing with mathematical models of a language in the course of their work, students of the humanities re­quiring a specific mathematical education, scientific workers dealing with any aspects whatsoever of cybernetics, etc.This book was written so that it could be used by readers who are not professional mathematicians. In any case, the basic material of the first five chapters are designed for such a reader. The sixth chapter requires some experience in reading mathematical literature. The seventh chapter is written especially for linguists and mathematicians dealing with mathematical linguistics. It is only a particular exam­ ple for the more general reader.Formally, the only prerequisites for reading this book are the knowledge of high school mathematics and a familiarity with certain elements of set theory (obtainable, for example, from Appendix 2). However, it would be helpful for the reader to possess an acquaintance with the elements of mathematics.The book contains 72 illus.About the author JULIUS SCHREIDERIs a docent and a Candidate of Physical and Mathematical Sciences. He is head of the sec­tor on semiotic problems of information at the All Union Institute of Scientific and Technical Information of the USSR Academy of Sciences. He is the author of one hundred and fifty pub­lished works, six of them books, including The Monte Carlo Method. Pergamon Press. 1966 (tran­slated from the 1962 Russian edition), papers on computer theory (Some of which have been translated into English), the semantic theory of information (Some of which have appeared in the journal Information Storage and Retrieval), ma­thematical linguistics, the mathematical theory of collectives, system theory (some of which have appeared in the journal Ideen der exakten Wissen. the 1973 and 1975 issues of the yearly "System Research”, etc.), the methodology of science ("Science—a Source of Knowledge and Superstition”. New World, No. 10, 1969. ‘‘Is Rea­ son Inherent in Computers?”, Questions of Philo­ sophy. No. 7. 1974).Julius Schreider’s publications have been tran­slated into English, German. Bulgarian. Polish, Hungarian. Japanese, Czech, Latvian and Ukrai­ nian. This book is also appearing in Hungarian and Polish translations.The book was translated from the Russian by Martin Greendlinger and was published by Mir in 1975.ContentsFrom the Introduction to the Russian Edition. 5 Preface. 8 List of Symbols. 11 Introduction. 13Chapter I. Relations. 16§ 1. How a Relation is Given. 16 § 2. Functions as Relations. 25 § 3. Operations on Relations. 29 § 4. Algebraic Properties of Operations. 37 § 5. Properties of Relations. 43 § 6. Invariance of Properties of Relations. 46Chapter II. Identity and Equivalence. 50§1. From Identity to Equivalence. 50 §2. Formal Properties of Equivalence. 57 §3. Operations on Equivalences. 65 § 4. Equivalence Relations on the Real Axis. 74Chapter III. Resemblance and Tolerance. 81§ 1. From Resemblance to Tolerance. 81 § 2. Operations on Tolerances. 94 § 3. Tolerance Classes. 95 § 4. A Further Exploration of the Structure of Tolerances. 107Chapter IV. Ordering 117§ 1. What is Order? 117 § 2. Operations on Order Relations. 135 § 3. Tree Orders. 142 § 4. Sets with Several Orders. 150Chapter V. Relations in School Mathematics. 159§ 1. RelationsBetween Geometric Objects. 159 § 2.Relations Between Equations. 163Chapter VI. Mappings of Relations 166§ 1. Homomorphisms and Correlations. 166 § 2. Minimal Image and Canonical Completion of a Relation. 171Chapter VII. Examples from Mathematical Linguistics 181§ 1. Syntactical Structures. 181 § 2. The General Concept of a Text. 202 § 3. Compatibility Models. 210§ 4. A Formal Problem in Decoding Theory. 218§ 5. On Distributions  222Appendix. 231§ 1. Summary of the Main Types of Relations and Their Properties. 231§ 2. Elementary Facts about Sets . 231§ 3. What is a Model? 245§ 4. Real Objects and Set-Theoretical Concepts. 250Index  275