Equality resemblance and order

Table of contents:


From the Introduction to the Russian Edition. 5

Preface. 8

List of Symbols. 11

Introduction. 13

Chapter 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. 46

Chapter 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. 74

Chapter 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. 107

Chapter IV. Ordering 117

1. What is Order? 117
2. Operations on Order Relations. 135
3. Tree Orders. 142
4. Sets with Several Orders. 150

Chapter V. Relations in School Mathematics. 159

1. Relations Between Geometric Objects. 159
2.Relations Between Equations. 163

Chapter VI. Mappings of Relations 166

1. Homomorphisms and Correlations. 166
2. Minimal Image and Canonical Completion of a Relation. 171

Chapter 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 222

Appendix. 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. 250

Index 275

This 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.