## relation

**-**

## Relation

# RELATION

Relation, from the Latin, *relatio* (*referre, relatum* ), means a reference, bearing, or towardness, and relative signifies the substantive meaning of something so ordered or referred. The Greeks devised a technical phrase for relation to emphasize the preposition "to" or "toward," viz, πρóς τι (Lat. *ad aliquid* ), which may be translated as "toward something." This article treats of the historical development of the concepts of relation and the relative, with emphasis on Greek, scholastic, and modern thought, and then offers a summary comparison of historical and doctrinal interest to thinkers in the scholastic tradition.

**Origins in Greek Thought.** The principal philosophers in the ancient Greek tradition who contributed to the development of the concepts of relation and the relative were plato, Aristotle, and plotinus.

*Plato.* In the *Sophist* (esp. 242B–254B) the Eleatic stranger and Theaetetus, personifying the materialist and idealist currents respectively, consider the former's conception of being as "the changing." Advancing this notion from what resists touch to what naturally possesses a power of acting or undergoing action, thence to power itself—taken for a distinctive sign of being (247B)—the dialectic finally introduces "soul" as a unifying principle (253B). Thus thought, as recognizing both movement and rest without becoming either, affords a principle resolving their opposition. This exposure of the three distinct genera—being, movement, and rest—when linked with the notions of action and passion, returns again to power, the sum of the properties outwardly expressive of the inward, incommunicable nature. Thought, then, provides a reach into something more than the isolated self-identity of the three genera of forms in their mutual distinctiveness and reciprocal relevance. From this emerge two more forms: selfhood or identity and nonselfhood or otherness. This last, grounded on power, is the basis for the Platonic teaching on relation.

The different applications of the πρóς (to, toward) of ordinary discourse, summed up in πρóς τι (toward something), are gradually unified in the *Sophist,* through the variant forms πρὸς ἄλλα and ἕτερου ἀεἰ πρὸς ἕτερον, in the distinct genus θάτερον (255C–D).

*Aristotle.* In the, *Categories,* book 1 of the *Organon,* Aristotle distinguishes τὰ πρός τι (relatives) as the third of his categories of being. A provisional (6a 36–37) and later revised (8a 32–33) definition of relatives sets forth the distinction between "being said of another" and "being actually related to another," which for Aristotle are not the same (8a 34). This is made more explicit in the *Metaphysics* (1020b 25–1021 b 11), where a studied treatment of the relative is given. Here a threefold division of relatives is proposed as complete: (1) that of the double to the half, the triple to the third, and, generally, the multiple to the multiplied and the container to the contained; (2) the "what can heat" to "what can be heated," the "what cuts" to "what is cut," and, in general, the active to the passive; (3) the measurable to the measure, the knowable to knowledge, and the sensible to sense.

The first are said "according to number" or "according to unity," the former giving the proportions derived from any continuous quantity, the latter the unities of substance (same), quantity (equal), and quality (similar). The second also is twofold, based on the distinction of potency and act: "what can heat" to "what can be heated," potential; "what heats" to "what is heated," acting. These vary also as to time: past, as father to son; future, as an action referred to what will be done. Even the incapable and the invisible are mentioned here.

The third mode troubles all later commentators, some even correcting the text to read "measure to measurable," since the parallels, "knowable" and "sensible," actually are what measure knowledge and sense. Further, whereas the *Categories* proposes all relatives as reciprocal, even using this as a critical premise (12b 20), in the *Metaphysics* Aristotle gives a distinct classification of πρός τι that explicitly excludes reciprocity.

Yet, since universal reciprocity is a demand for correlative terms (not mutual relationship), one can hold that in treating of relative terms Aristotle is attending mainly to one aspect of the measurable, the knowable, and the sensible—themselves peculiar terms in that each is so named not from anything intrinsic to it but from an external referent: a dependency in being, relatively named. The other two classes furnish extremes, each incorporating its own entitlement to being relatively named. "Measurable," by naming only a terminal function for another's reference, viz, the measure's, becomes a distinctive relative not signifying like the bilaterally related examples of the other two sets.

*Plotinus.* In his sixth Ennead, tract. 1–3, Plotinus refutes the Aristotelian genera by attacking their univocities, and exposes the equivocity of relation by applying it beyond any single category. But carrying the Platonic analysis toward Aristotle, he further fixes motion between action and passion, linking the power of Plato with the potency of Aristotle: "…arts (like boxing) as dispositions of the soul are qualities, as outwardly orientated they are active, and as directed to an outside object they are relative" (1.12). He also relegates being to the "higher realm" and lists becoming in the categories, thereby reducing the whole "lower realm" to two genera, one comprising matter, form, and composite, and the other relation (3.3).

**Scholastic Development.** Among the scholastics, relation is generally first distinguished as a distinct category of accident, having its being in a subject not absolutely but in respect to another. This they call "predicamental" relation. Then, confronted with other relations no less real but not fitting this accidental category (e.g., means, appetite, creation), they named these "transcendental" in contrast. Still other relations, abundantly used but not explicitly classified by Aristotle (e.g., sign, universal, predicate), remain, and these were standardized as "relations of reason." The last include all nonreal or logical relations, and thus are distinguished from "real" relations, which embrace both the predicamental and the transcendental.

*Transcendental Relations.* Christian philosophers, exposed to cases of real relation that could not be classified as accidental, as in the Holy trinity and creation, enumerated such transpredicamental relations as transcendental. Either they anchored these in tradition as the third mode identified in Aristotle's *Metaphysics,* or they based their distinction on the expression *relativum secundum dici vel secundum esse* (relative as to speech or as to reality), which was formulated from Aristotle's double definition, but has been abusively wielded to the present day despite the teaching of St. Thomas Aquinas (*De pot.* 7.10. ad 11). A typical, scholastic manualist definition of this relation calls it "the order included in some absolute essence" [J. Gredt, *Elementa Philosophiae Aristotelico-Thomisticae* (Freiburg 1937) 1:154–155].

*Predicamental Relations.* Predicamental relations, whether taken as real *secundum esse* or as the first two Aristotelian modes based on quantity or on action and passion, are defined as real accidents, distinct from substance, whose whole nature consists in reference to another. Beyond this definition, however, little unanimity obtains.

*Foundations of Relations.* Most divergencies stem from basic tenets regarding the foundation of relations and the distinction between relations and their foundations. Classical thomism distinguishes the relation itself from the extremes (subject and term) and the foundation, sometimes reducing extremes and foundation to the material and formal causes of the reference constituting the relation.

If relations are really distinct from their foundations, then differences in foundation provide causal difference for the kinds of relation; but if relations are really no more than the underlying absolutes, merely viewed and hence named relatively, then relatedness is a being of reason (logical) and the supporting essence the only real entity. On such a basis nominalists generally reject the "ancient" predicamental relation and call the foundation "transcendental relation" (see nominalism).

Persuaded of the real identity of relation with its foundation, F. suÁrez is ranged rather with the nominalists and therefore unable to see how St. Thomas or duns scotus can distinguish predicamental and transcendental relations (*Meta. disp.* 2.37.2). (see suarezianism).

**Relation in the Modern Period.** For most modern thinkers, theories of relation were reducible to the problem of knowledge of relation, as a result of the dismissal of substance and causality from true objectivity. The resulting schools may be conveniently classified under the titles of empiricism, criticism, and idealism (see criticism, philosophical).

*Empiricism.* The empiricism of John locke, reducing knowledge of substance to knowledge of qualities, distinguishes primary and secondary qualities, with only the primary ones known surely, but in a general way, and without any discoverable connections between them and secondary qualities. By making the generality of an idea a function of the second operation of the mind, "relating," Locke reduces signification to nothing but a relation added by mind. Knowledge, defined as only "perception of the connexion and agreement, or disagreement and repugnancy of any of our ideas," thus shifts from the idea to the relation between ideas. Types of judgment then follow the various relations of agreement or disagreement, yielding two further divisions distinguished as mental and concrete connections. (see knowledge, theories of.)

These last, ideal relations and relations among matters of fact, become the "invariable" and "variable" relations of hume, with only the former rendering any strict knowledge, since they alone are purely ideal. Invariable relations are twofold: those belonging to intuition (qualitative), and those to demonstration (quantitative). This makes mathematics the only strict science, and leaves to fallible "moral reasoning" the existential world of matters of fact, thus effecting a dichotomy between ideal demonstration and experimental belief.

*Criticism.* Kant, marking the two sides of this chasm as the *ratio veritatis* (explanation of ideal essence) and the *ratio existendi vel actualitatis* (fact of a thing's reality), distinguishes analytic method from analytic judgment, and the order of logical relations of ideas from the order of relevance of these to actual relatives. Because Kant holds that up-down and near-far are not really relations between parts of reality, but to us and through us are related to some absolute frame, he is led to space and time as forms of the mind, coordinates for the singular and sensible. Universal concepts of the understanding, on the other hand, have no such frames; hence, for him, there are two irreducible kinds of knowledge. Avoiding Hume's path to skepticism, he is led to look to a kind of knowledge independent of experience, hence to pure knowledge, thence to pure reason.

Reversing the relationship of mind and thing, making the former measure rather than measured, Kant fixes the a priori forms of his categories. These, taken either absolutely or relatively, "unschematicized" or "schematicized" by imagination to relate to sense intuition, form the basis for the distinction between phenomenal and noumenal concepts. His list of categories includes quantity, quality, relation, and modality, with relation including substantiality, causality, and interaction. The corresponding judgments show "all the relations of thought in judgment" gathered under relations of predicate-subject (categorical), cause-effect (hypothetical), and of submembership. But instead of the traditional predicate as related to subject, this transcendental method now centers on the pure relations between, whether these be analytic (regressive) or synthetic (progressive).

Following that of Kant, the modern emphasis remains on the relations rather than on the terms treated as correlatives, becoming ever more pronounced in both modern physics and modern mathematics, where the old primacy of substance yields to function, and logic becomes a calculus of relations.

*Idealism.* schelling declares that both dogmatism (Descartes through Spinoza) and criticism (Kant through Fichte) failed in their attempts: the former to reduce subject to absolute object, the latter to reduce object to absolute subject. Favoring criticism, his solution dissolves the distinction of both subject and object in the absolute. How Schelling's theory of relation directs this may be seen from the first and last of his five metaphysical principles: the law of identity, based on the ubiquitous bond of ground and consequent; and the law of reciprocity, ruling that every nature is revealed in its opposite. The underlying unity of opposition should resolve by ever diminishing distinction into the identity of the absolute. A dynamic pantheism results from the two logical functions of the first principle: every relation between ground and consequent produces a predication of identity (so the pantheism); but also conversely, every predication of identity discloses, then, a process from ground to consequent (hence, the emanationism or dynamism).

hegel is disappointed with this indifferent absolute: "calling a cow black because seen at night." For him, three steps involving relation escape the abstractness and vacuity of ordinary universals: (1) the "concrete universals," grasping mutual relations, express the unity in difference and the interrelatedness of actual things; (2) the roles of "negativity" and "mediation" determine otherness, negation exposing relations to what an idea is not (supplementing its abstractness and making it concrete), while mediation relates it to all other points of reference for systematic connections; (3) the positive result by negation-of-negation comes through the triadic dialectic: thesis, first stage of the concrete universal, which negated gives antithesis, which in turn negated completes the mediation in synthesis, itself a new thesis for further development. Hegel sees this system of organic relations of concretion as justified in that this is just the way one must expect Absolute Spirit to be evolving if It is self-developing.

**Relation in Logic: A Comparison.** The whole of Aristotelian logic concerns relations. Taking "intention" for relation, the intellect's registration of reality can be viewed as "intentional," related to reality as measured to measure. This is St. Thomas's view of the material and formal aspects presented by logical relations. "First intentions" are the (real) relations whereby man variously engages the real. "Second intentions" arise from the sub-sequent mutual relevance of these concepts for their distinction and ordering: genus-species, subject-predicate, *etc.* (see intentionality; predicables; concept.)

logic is both the art of reasoning and the science of second intentions. Rules controlling the intentionalities of reasoning from the formal side are easily mastered as an art; the rules for integrating second intentions with their material ground in first intentions pertain to the more advanced consideration of the science. The first without the other leads to a mere formal rationalism, as history sadly attests; but the latter, fully taken, is so taxing as to be commonly forfeited.

What Hume did to experience (reducing it to relations of ideas), and Kant to the notion of substance (subsuming it under his category of relation) frustrates any retrace of logical intentions to real intentions, thus transforming the nature of logic. If judgments and their expression in propositions be concerned only with the interlinking relations, neglecting the terms, as Locke held, relationality rather than intentionality becomes the new subject of logic. Such relations are commonly distinguished according to symmetry, transitivity, and reflexivity. Symmetrical and asymmetrical differ as the simultaneous and the prior; transitive and intransitive differ as "on the right" and "next to." Rules of inference treat symmetrical statements as convertible (*A* is simultaneous to *B, B* is simultaneous to *A* ), and explore the consequences of transitivity and intransitivity as expounded by William james and systematized by Boole, de mor gan, russell, whitehead, et al.

An adaptation of modern symbols as a shorthand for traditional consequences is often treated in scholastic manuals; however, such algebraic manipulation of symbols seems inadequate by itself to capture and control the intricate causalities required for philosophical demonstration. Although considerable advances have thereby been made in the area of formal logic, and therefore in dialectic, the material logic of demonstration remains concerned with a real, causal level that is itself irrelevant to this relational calculus.

**Critique and Evaluation.** Plato's identification of the other and its reciprocity with the same or self opens a wedge between his absolutes, which Aristotle notes in commenting that if all the Ideas were absolutes, the idea of relation would itself be repugnant (*Meta.* 991b 15–18). Aristotle's mind cannot be measured by the *Categories* alone. His *Metaphysics,* treating relatives (not relations as such), distinguishes correlatives named for a relatedness from their own standpoint, from others named only because of a relatedness found in their opposite: thus confounding all the commentators who confuse "relation" with the "relative." Plotinus, reverting to Forms, fails to see merit in the Aristotelian categories; but his attack on the univocity of relation highlights its analogous nature, a source of later difficulties and solutions.

*Thomistic Doctrine on Relation.* Among the scholastics, St. Thomas alone seems to emend Aristotle's teaching with precision. His capital texts, the *Commentary on the Metaphysics* and the *De potentia,* qq. 7–8, give a fully developed doctrine of both relatives and relation. Composite substance—that analyzed through the categories—furnishes two basic accidents, quantity (from its matter) and quality (from its form), twin sources of all real relations, which also underlie the causality of action and passion. His reduction of all the real to these two fonts alone is constant, and almost invariably he invokes the authority of the *Metaphysics.* But he does not exclude real relation from the third mode (nor does he exclude logical relations from the others). Most important is his postponement of any such question until first having indicated the unique relative that is wholly indebted to its correlative even for being named relatively. Whatever relatedness this correlative involves must be reducible, precisely as real, to the real causality founding the other two modes, but now taken transcendentally—i.e., not limited to the univocal senses of the categories, where action means transitive action and quantity means corporeal dimensiveness. That action, for instance, admits of transpredicamental extension by analogical transference is commonly understood; such relations simply follow this transference. And such correlatives, always in diverse genera or orders, disclose the root of their onesidedness in their basic dependence, itself often springing from a causal reference that makes them real.

*The Relatedness of Knowledge.* The problem of the knowledge of relation that engages modern thinkers as epistemology is, for St. Thomas, essentially the relation of knowledge, an instance of measurable to measure. The very name and notion of object demand this specification of the immanent activity of knowledge by some external reality. Crucial to the critical problem, therefore, is the totality of progressively immanent causality from the sensed to its being understood. For all of this the "third mode" alone, since committed to a diversity of orders, hardly suffices. Antecedent causalities in action and passion, and even in quantity, integrate the Thomistic analysis, making a circuit from the physicalphysiological immutation by sense qualities, through the relational transition to perceptual-spiritual immutation of the internal senses, to the intellect's reflexive attainment of this objective content *with the relations themselves.* (see knowledge, process of.)

The correlation of the modern primary and secondary sense qualities (the common and proper sensibles) with relatives of the first two modes, and these with the internal and external senses, is thus prerequisite to any examination of comprehension by the intellect.

*Relation and Analogy.* But not only is the study of relatives prior to that of relation; the search for relations also exceeds any limited presentation in terms of "relative terms." Besides the relationship designated by every preposition, and the order and reference signified by relative terms, sometimes words are intentionally made relational to encompass some observable unity while still allowing for variation.

Such is the nature of analogical signification, basically the naming of correlatives by an identical term to signify their relational community. Denying such a relationship means either taking the name equivocally for the several applications (allowing for no causal connection) or taking it univocally (allowing for no oppositional distinction). Further attention to relations between relations (or proportions of proportions) results in "proportionality," an analogy reducible to the sharing of a common name on the part of comparable extremes of diverse sets because of relatedness within the sets. (see analogy.)

A further stage of analogical import arises in inferences drawn from this relational interplay between elements in proportionality. Because *a:b::c:d* (genus is to difference as matter is to form), then by permutation or transumption, *a:c::b:d:* (genus is to matter as difference is to form). St. Thomas frequently uses this dialectical device to set up parallels like the following: just as genus is taken from the matter of the subject, so difference derives from the form, thereby exposing the underlying causality that justifies the permutation.

*Current Positions.* Later scholastics, lacking critical knowledge of Thomistic texts, seem to have passed on something less. And the modern manualists claiming this tradition usually constrain themselves by their method to excluding necessary but delicate analogies. Their position on relation generally ruins their doctrine of analogy, thus reciprocally spoiling their doctrine on relation also.

Classical modern thinkers, rejecting the Wolffian version of scholasticism, sacrificed centuries of attainment to make new starts. But starts that stop short of substance or cause, intelligence or sense, can find only rational relations and logical analogies disassociated from the real, and thus futile for philosophy. This heritage prevails in the current developments of logical positivism, as present problems in personalism keep alive Plato's "self" and "other." Enrichment awaits all in the doctrines of person, knowledge, and analogy, as properly explained through the Thomistic notion of relation. But the warning of history is here: not only more, but different doctrine is available in masters like Aristotle and St. Thomas; it is hazardous to substitute their simplifiers.

See Also:relations, trinitarian; relativism; order; accident; similarity.

**Bibliography:** v. mathieu, *Enciclopedia filosofica*, 4 v. (Venice-Rome 1957) 4:26–41. a. krempel, *La Doctrine de la relation chez saint Thomas* (Paris 1952) reviewed by m. a. philippe in *Bulletin Thomiste* 9 (1955) 363–69 and *Revue des sciences philosophiques et théologiques* 42 (1948) 265–75. m. j. adler and r. m. hutchins, eds., *The Great Ideas,* 2 v. (Great Books of the Western World 2–3; Chicago 1952). r. eisler, *Wörterbuch der philosophischen Begriffe*, 3 v. (4th ed. Berlin 1927–30) 2:668–87.

[b. mattingly]

## relation

**relation** •**ashen**, fashion, passion, ration
•**abstraction**, action, attraction, benefaction, compaction, contraction, counteraction, diffraction, enaction, exaction, extraction, faction, fraction, interaction, liquefaction, malefaction, petrifaction, proaction, protraction, putrefaction, redaction, retroaction, satisfaction, stupefaction, subtraction, traction, transaction, tumefaction, vitrifaction
•**expansion**, mansion, scansion, stanchion
•sanction
•**caption**, contraption
•**harshen**, Martian
•**cession**, discretion, freshen, session
•**abjection**, affection, circumspection, collection, complexion, confection, connection, convection, correction, defection, deflection, dejection, detection, direction, ejection, election, erection, genuflection, imperfection, infection, inflection, injection, inspection, insurrection, interconnection, interjection, intersection, introspection, lection, misdirection, objection, perfection, predilection, projection, protection, refection, reflection, rejection, resurrection, retrospection, section, selection, subjection, transection, vivisection
•**exemption**, pre-emption, redemption
•**abstention**, apprehension, ascension, attention, circumvention, comprehension, condescension, contention, contravention, convention, declension, detention, dimension, dissension, extension, gentian, hypertension, hypotension, intention, intervention, invention, mention, misapprehension, obtention, pension, prehension, prevention, recension, retention, subvention, supervention, suspension, tension
•**conception**, contraception, deception, exception, inception, interception, misconception, perception, reception
•Übermenschen • subsection
•**ablation**, aeration, agnation, Alsatian, Amerasian, Asian, aviation, cetacean, citation, conation, creation, Croatian, crustacean, curation, Dalmatian, delation, dilation, donation, duration, elation, fixation, Galatian, gyration, Haitian, halation, Horatian, ideation, illation, lavation, legation, libation, location, lunation, mutation, natation, nation, negation, notation, nutation, oblation, oration, ovation, potation, relation, rogation, rotation, Sarmatian, sedation, Serbo-Croatian, station, taxation, Thracian, vacation, vexation, vocation, zonation
•**accretion**, Capetian, completion, concretion, deletion, depletion, Diocletian, excretion, Grecian, Helvetian, repletion, Rhodesian, secretion, suppletion, Tahitian, venetian
•**academician**, addition, aesthetician (*US* esthetician), ambition, audition, beautician, clinician, coition, cosmetician, diagnostician, dialectician, dietitian, Domitian, edition, electrician, emission, fission, fruition, Hermitian, ignition, linguistician, logician, magician, mathematician, Mauritian, mechanician, metaphysician, mission, monition, mortician, munition, musician, obstetrician, omission, optician, paediatrician (*US* pediatrician), patrician, petition, Phoenician, physician, politician, position, rhetorician, sedition, statistician, suspicion, tactician, technician, theoretician, Titian, tuition, volition
•**addiction**, affliction, benediction, constriction, conviction, crucifixion, depiction, dereliction, diction, eviction, fiction, friction, infliction, interdiction, jurisdiction, malediction, restriction, transfixion, valediction
•**distinction**, extinction, intinction
•**ascription**, circumscription, conscription, decryption, description, Egyptian, encryption, inscription, misdescription, prescription, subscription, superscription, transcription
•proscription
•**concoction**, decoction
•**adoption**, option
•**abortion**, apportion, caution, contortion, distortion, extortion, portion, proportion, retortion, torsion
•auction
•**absorption**, sorption
•**commotion**, devotion, emotion, groschen, Laotian, locomotion, lotion, motion, notion, Nova Scotian, ocean, potion, promotion
•**ablution**, absolution, allocution, attribution, circumlocution, circumvolution, Confucian, constitution, contribution, convolution, counter-revolution, destitution, dilution, diminution, distribution, electrocution, elocution, evolution, execution, institution, interlocution, irresolution, Lilliputian, locution, perlocution, persecution, pollution, prosecution, prostitution, restitution, retribution, Rosicrucian, solution, substitution, volution
•cushion • resumption • München
•pincushion
•**Belorussian**, Prussian, Russian
•**abduction**, conduction, construction, deduction, destruction, eduction, effluxion, induction, instruction, introduction, misconstruction, obstruction, production, reduction, ruction, seduction, suction, underproduction
•**avulsion**, compulsion, convulsion, emulsion, expulsion, impulsion, propulsion, repulsion, revulsion
•**assumption**, consumption, gumption, presumption
•**luncheon**, scuncheon, truncheon
•**compunction**, conjunction, dysfunction, expunction, function, junction, malfunction, multifunction, unction
•**abruption**, corruption, disruption, eruption, interruption
•T-junction • liposuction
•**animadversion**, aspersion, assertion, aversion, Cistercian, coercion, conversion, desertion, disconcertion, dispersion, diversion, emersion, excursion, exertion, extroversion, immersion, incursion, insertion, interspersion, introversion, Persian, perversion, submersion, subversion, tertian, version
•excerption

## Relation

# Relation

In mathematics, a relation is any collection of ordered pairs. The fact that the pairs are ordered is important, and means that the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. For most useful relations, the elements of the ordered pairs are naturally associated or related in some way.

More formally, a relation is a subset (a partial collection) of the set of all possible ordered pairs (a, b) where the first element of each ordered pair is taken from one set (call it A), and the second element

### KEY TERMS

**Cartesian product—** The Cartesian product of two sets A and B is the set of all possible ordered pairs (a, b) formed by taking the first element of the pair from the set A and the second element of the pair from the set B.

**Domain—** The set of elements appearing as first members in the ordered pairs of a relation.

**Function—** A function is a relation for which no two ordered pairs have the same first element.

**Ordered pair—** An ordered pair (a, b) is a pair of elements associated in such a way that order matters. That is, the ordered pair (a, b) is different from (b, a) unless a = b.

**Range—** The set containing all the values of the function.

**Set—** A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers.

**Subset—** A set, S, is called a subset of another set, I, if every member of S is contained in I.

of each ordered pair is taken from a second set (call it B). A and B are often the same set; that is, A = B is common. The set of all such ordered pairs formed by taking the first element from the set A and the second element from the set B is called the Cartesian product of the sets A and B, and is written A× B. A relation between two sets then, is a specific subset of the Cartesian product of the two sets.

Since relations are sets at ordered pairs they can be graphed on the ordinary coordinate plane if they have ordered pairs of real numbers as their elements (real numbers are all of the terminating, repeating and non-repeating decimals); for example, the relation that consists of ordered pairs (x, y) such that x = y is a subset of the plane, specifically, those points on the line x = y (Figure 1). Another example of a relation between real numbers is the set of ordered pairs (x, y), such that x> y. This is also a subset of the coordinate plane, the half-plane below and to the right of the line x = y, not including the points on the line (Figure 2). Notice that because a relation is a subset of all possible ordered pairs (a, b), some members of the set A may not appear in any of the ordered pairs of a particular relation. Likewise, some members of the set B may not appear in any ordered pairs of the relation. The collection of all those members of the set A that appear in

at least one ordered pair of a relation form a subset of A called the domain of the relation. The collection of members from the set B that appear in at least one ordered pair of the relation form a subset of B called the range of the relation. Elements in the range of a relation are called values of the relation. One special and useful type of relation, called a function, is very important. For every ordered pair (a, b) in a relation, if every a is associated with one and only one b, then the relation is a function. That is, a function is a relation for which no two of the ordered pairs have the same first element. Relations and functions of all sorts are important in every branch of science, because they are mathematical expressions of the physical relationships we observe in nature.

## Resources

### BOOKS

Bittinger, Marvin L., and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

McKeague, Charles P. *Intermediate Algebra.* 5th ed. Stamford, CN: Brooks/Cole Publishing, 2002.

Swan, Paul. *Patterns in Mathematics.* Rowley, MA: Didax Educational Resources: Rev. ed., 2003.

J. R. Maddocks

## Relation

# Relation

In **mathematics** , a relation is any collection of ordered pairs. The fact that the pairs are ordered is important, and means that the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. For most useful relations, the elements of the ordered pairs are naturally associated or related in some way.

More formally, a relation is a subset (a partial collection) of the set of all possible ordered pairs (a, b) where the first element of each ordered pair is taken from one set (call it A), and the second element of each ordered pair is taken from a second set (call it B). A and B are often the same set; that is, A = B is common. The set of all such ordered pairs formed by taking the first element from the set A and the second element from the set B is called the Cartesian product of the sets A and B, and is written A × B. A relation between two sets then, is a specific subset of the Cartesian product of the two sets.

Since relations are sets at ordered pairs they can be graphed on the ordinary coordinate **plane** if they have ordered pairs of **real numbers** as their elements (real numbers are all of the terminating, repeating and nonrepeating decimals); for example, the relation that consists of ordered pairs (x, y) such that x = y is a subset of the plane, specifically, those points on the line x = y. Another example of a relation between real numbers is
the set of ordered pairs (x, y), such that x > y. This is also a subset of the coordinate plane, the half-plane below and to the right of the line x = y, not including the points on the line. Notice that because a relation is a subset of all possible ordered pairs (a, b), some members of the set A may not appear in any of the ordered pairs of a particular relation. Likewise, some members of the set B may not appear in any ordered pairs of the relation. The collection of all those members of the set A that appear in at least one ordered pair of a relation form a subset of A called the **domain** of the relation. The collection of members from the set B that appear in at least one ordered pair of the relation form a subset of B called the range of the relation. Elements in the range of a relation are called values of the relation. One special and useful type of relation, called a **function** , is very important. For every ordered pair (a, b) in a relation, if every a is associated with one and only one b, then the relation is a function. That is, a function is a relation for which no two of the ordered pairs have the same first element. Relations and functions of all sorts are important in every branch of science, because they are mathematical expressions of the physical relationships we observe in nature.

## Resources

### books

Bittinger, Marvin L., and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Kyle, James. *Mathematics Unraveled.* Blue Ridge Summit, PA: Tab Books, 1976.

McKeague, Charles P. *Intermediate Algebra.* 5th ed. Fort Worth: Saunders College Publishing, 1995.

J. R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Cartesian product**—The Cartesian product of two sets A and B is the set of all possible ordered pairs (a, b) formed by taking the first element of the pair from the set A and the second element of the pair from the set B.

**Domain**—The set of elements appearing as first members in the ordered pairs of a relation.

**Function**—A function is a relation for which no two ordered pairs have the same first element.

**Ordered pair**—An ordered pair (a, b) is a pair of elements associated in such a way that order matters. That is, the ordered pair (a, b) is different from (b, a) unless a = b.

**Range**—The set containing all the values of the function.

**Set**—A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers.

**Subset**—A set, S, is called a subset of another set, I, if every member of S is contained in I.

## relation

**relation** (defined on sets *S*_{1}, *S*_{2},… *S _{n}*) A subset

*R*of the Cartesian product

*S*

_{1}×

*S*

_{2}× … ×

*S*

_{n}of the

*n*sets

*S*

_{1}, …,

*S*. This is called an

_{n}*n*-ary relation. When a relation

*R*is defined on a single set

*S*the implication is that

*R*is a subset of

*S*×

*S*× … ×

*S*(

*n*terms)

The most common situation occurs when

*n*= 2, i.e.

*R*is a subset of

*S*

_{1}×

*S*

_{2}. Then

*R*is called a

*binary relation*on

*S*

_{1}to

*S*

_{2}or between

*S*

_{1}and

*S*

_{2}.

*S*

_{1}is the

*domain*of

*R*and

*S*

_{2}the

*codomain*of

*R*. If the ordered pair (

*s*

_{1},

*s*

_{2}) belongs to the subset

*R*, a notation such as

*s*

_{1}

*R s*

_{2}or

*s*

_{1}ρ

*s*

_{2}

is usually adopted and it is then possible to talk about the relation

*R*or ρ and to say that

*s*

_{1}and

*s*

_{2}are related.

An example of a binary relation is the usual “is less than” relation defined on integers, where the subset

*R*consists of ordered pairs such as (4,5); it is however more natural to write 4 < 5. Other examples include: “is equal to” defined on strings, say; “is the square root of” defined on the nonnegative reals; “is defined in terms of” defined on the set of subroutines within a particular program; “is before in the queue” defined on the set of jobs awaiting execution at a particular time.

The function is a special kind of relation. Graphs are often used to provide a convenient pictorial representation of a relation.

Relations play an important part in theoretical aspects of many areas of computing, including the mathematical foundations of the subject, databases, compiling techniques, and operating systems. See also equivalence relation, partial ordering.

## relation

re·la·tion
/ riˈlāshən/
•
n.
1.
the way in which two or more concepts, objects, or people are connected; a thing's effect on or relevance to another:
*questions about the relation between writing and reality*

*the size of the targets*∎ (relations) the way in which two or more people, countries, or organizations feel about and behave toward each other:

**bore**no**relation to**their importance.*the improvement in relations between the two countries*

*the meetings helped cement Anglo-American relations.*∎ (relations) chiefly formal sexual intercourse:

*he wanted an excuse to abandon sexual relations with her.*2. a person who is connected by blood or marriage; a kinsman or kinswoman:

*she was no relation at all, but he called her Aunt Nora.*3. the action of telling a story. PHRASES: in relation to in the context of; in connection with:

*there is an ambiguity in the provisions in relation to children's hearings.*

## Relation

# RELATION

*Kin; relative. The connection of two individuals, or their situation with respect to each other, who are associated, either by law, agreement, or kinship in a social status or union for purposes of domestic life, such as*parent and child*or*husband and wife.

The doctrine of *relation* is the principle by which an act performed at one time is deemed, through a legal fiction, to have been performed at a prior time. For example, in the conveyance of real property, the final proceeding that completes the transfer of property is considered, for certain purposes, to have become effective by relation as of the day when the first proceeding took place. Relation, in essence, is the legal term for retroactive effect.