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a philosophical category that expresses the nature of the disposition of the elements of a particular system and their interdependence; an individual’s emotional-volitional attitude toward something, the expression of his position; and the mental comparison of different objects or aspects of a given object.
Dialectical materialism postulates that a relation has an objective and universal character. Only things, their properties, and relations that are interconnected and interrelated in infinitely many ways with other things and properties exist in the world. V. I. Lenin regarded as accurate Hegel’s idea that each concrete thing stands in various relations to all other things (Poln. sobr. soch., 5th ed., vol. 29, p. 124). Relations form systems of varying degrees of complexity from suitable elements, and a given relation may be present in various things (internal relations) or between different things (external relations). An example is any law as a significant relation between things and phenomena. Conversely, a given thing can enter into infinitely diverse relations with other things, which characterizes the multiplicity of the properties of a given thing. Anything may be considered as the relationship among its constituent elements, and if the relationship changes the thing itself also changes. For example, a different arrangement of the same elements in the words kot (cat) and tok (current) changes the meaning of the words. At the same time, any relation characterizes precisely those things between which it exists. For example, the relation “less than” or “greater than” characterizes quantities; the relation “south of describes the position of something with respect to something else; and the relation “father” denotes kinship. Consequently, a relation may function as a property or attribute of things. In different relations a thing reveals different and even contradictory properties. The relations of objects and phenomena with respect to each other are infinitely diverse and include spatial, temporal, and causal relations, the relations between part and whole, and the relations between form and content and between the external and internal. Social relations constitute a special type of relation.
Scientific thinking reveals the essence of things and the lawlike regularities of their origin and development by identifying their relations with other things. Describing the elements of the dialectic, Lenin pointed out the necessity of investigating relations: “The entire totality of the manifold relations of this thing to others,” “the relations of each thing … are not only manifold, but general, universal. Each thing (phenomenon, process, …) is connected with every other; the endless process of the discovery of new sides, relations” (ibid., pp. 202–03). The growing role of systems-structural methods of investigation has made the category of relations increasingly important in modern science.
A. G. SPIRKIN
Relations in logic. In the meaningful statements of natural languages, a relation is usually expressed by a predicate that has more than one subject or one subject with complements. Depending on the number of subjects (and complements), they are called the terms, subjects, or elements of the given relation. A distinction is made between two-place (binary, two-term) relations (“a is less than b,” “the Oka is shorter than the Volga,” “the rails are parallel to each other”), three-place relations (ternary, three-term; “point A lies between B and C,” “2 plus 3 equals 5”), four-place relations (“the numbers x1, y1, x2, and y2 are proportional”), and n-place (n-ary, n-term) relations. These meaningful ideas are realized in the precise terminology of set theory (algebra) and mathematical logic. The first of these precise reformulations reflects the extensional aspect of the concept of relation, and the second reflects the intensional (meaningful, contensive).
In set-theoretic terminology, by a binary (n-ary) relation we mean a set of ordered pairs (respectively, ordered n-tuples) of the members of some set (the field of the given relation). If the ordered pair < x, y > belongs to some relation R, then it also is said that x stands in relation R to y [symbolically, R(xy) or xRy]. The set of the first elements of ordered pairs belonging to relation R is its domain of definition (departure), and the set of the second elements constitutes the range of values (arrival). Similar concepts also are introduced for multiplace relations. A relation consisting of the pairs < y, x > obtained by transposition of the members of a given relation R of the pairs < x, y > is said to be the inverse of R and is designated as R-1. The range of values of one of these mutually inverse relations [the term is justified by the fact that (R-1)-1 is always equal to R] serves as the domain of definition of the other, and the domain of definition serves as the range of values. Since relations are particular cases of sets, the operations of set theory, in particular the union, intersection, and complement of relations, are introduced for them in the usual manner. Let us consider certain properties and basic types of binary relations—the class of relations that is most important for application and theoretical constructions.
PROPERTIES OF BINARY RELATIONS. If for any x, xRx is true, then R is called reflexive; examples include the relation of the equality of numbers (every number is equal to itself) and the similitude of triangles. If for every x, xRx does not obtain (symbolically, ⌉ xRx), then R is said to be antireflexive, or irreflexive (for example, the relation of the perpendicularity of straight lines —no straight line is perpendicular to itself)- If for any x and y that are not equal to each other, one stands in relation R to the other, that is, if one of the three relations xRy, x = y, or yRx is satisfied, then R is said to be connected (for example, the relation <). If for any x and y, yRx follows from xRy, then R is said to be symmetric (for example, the relation of equality = or the relation of inequality ≠). If for any x and y, x = y follows from xRy and xR-1y (that is, if R and R-1 are satisfied simultaneously only for equal terms), then R is called antisymmetric (for example, the relation ≤ or ≥ for any objects). If for any x and y, ⌉ xRy follows from xRy, then R is called asymmetric (such as the relation < or >, since no object is greater or smaller than itself). If for any x, y, and z, xRz follows from xRy and yRz, then R is said to be transitive (for example, the relations = or <, but not ≠). It also would be possible to determine other properties of binary relations, but it is not difficult to show that all other properties are defined in terms of these properties by means of logical operations.
TYPES OF RELATIONS. Many of the types of relations cited below have already been encountered in the examples given above. Combining the properties of reflexivity, symmetry, and transitiveness, we come to the most important type of relation —the relation of equality (identity, equivalence). It is not difficult to show that any such relation induces (defines) a partitioning of the set in which it is defined into nonintersecting classes, called equivalence classes: the elements connected by a given relation fall into a common class, whereas those that are not connected fall into different classes. Thus, elements that fall into a common class are in some sense indistinguishable, which determines the importance of this type of relation.
REFERENCESTarski, A. V vedenie v logiku i metodologiiu deduktivnykh nauk. Moscow, 1948. (Translated from English.)
Church, A. Vvedenie v matematicheskuiu logiku, vol. 1. Moscow, 1960. (Translated from English.)
Uemov, A. I. Veshchi, svoistva i otnosheniia. Moscow, 1963.
Shreider, Iu. A. Ravenstvo, skhodstvo, poriadok. Moscow, 1971.
IU. A. GASTEV
See equivalence relation, partial ordering, pre-order, total ordering.