*71. ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *71.
In this number we shall be concerned with the more elementary properties of one-many, many-one, and one-one relations. These properties are very numerous and very important. The properties of many-one relations (i.e. of relations belonging to the class ) result from those of one-many relations by means of *70·5, whence it follows that many-one relations are the converses of one-many relations. It is thus only necessary to interchange and , and , and in order to obtain a property of a many-one relation from a property of a one-many relation. Or we may repeat the various steps of any proof, making the above interchanges at every step, and the analogous proposition will result. For this reason, in what follows, we shall omit all proofs of properties of many-one relations, confining ourselves to proving the analogous properties of one-many relations.
In virtue of *70·42, one-one relations (i.e. relations belonging to the class ) are the relations which are both one-many and many-one; hence their properties result from combining the properties of one-many and many-one relations. We shall omit the proofs when they consist merely in such combinations.
A one-many relation gives rise to a descriptive function which is existent whenever its argument belongs to the converse domain of the relation. That is, if , we have whenever . Conversely, if a descriptive function exists for the argument , then is one-many so far as that argument is concerned, i.e. . Thus we find
The descriptive function derived from a one-many relation has thus a definite value whenever , and not otherwise. Thus the class of arguments for which such a function exists is the converse domain of the relation which gives rise to the function, i.e. and the converse implication also holds.
It often happens that a relation which is not in general one-many becomes so when its domain, converse domain, or field is subjected to some limitation. For example, let be the relation of parent to child, the class of males, and the class of females. Then is not one-many, but and are one-many, and in fact ( = the father of , ( = the mother of . We shall often have occasion to deal with relations obtained by limitations imposed on or ; thus belongs to the class , and has for its domain. The class may be so constituted that only one relation fulfils this condition; in that case, . Since , we find . This type of condition, or or , is one which frequently occurs in subsequent work. Another condition which often occurs is . When this condition is realized, a term which belongs to the field of one relation of the class does not belong to the field of any other relation of this class, i.e. the fields of relations of this class are mutually exclusive.
For purposes of realizing imaginatively the properties of one-many relations, it is often convenient to picture their structure as in the accompanying figure.
Here , , , ... form the domain of , and all the points in the oval marked are such that has the relation to each of them, with similar conditions for and . What characterizes as a is the absence of overlapping in the ovals. For if and had a point in common, this would be a relatum both to and , and both and would be referents to it; whereas in a , no term has more than one referent.
The above figure illustrates a very important property of one-many relations, namely
In the above figure, is the relation of identity confined to , , ,.... If were not a , we could sometimes go from to some term of by the relation , and thence back to by the relation . But when , must bring us back to the point from which we started.
When , each of the ovals , , , ... in the above figure shrinks to a single point, so that . Thus when is given as a , it will be a if . This proposition is constantly used, and so is the consequence that is a if . (These propositions are *71·54·55 below.)
The hypothesis is equivalent to the hypothesis (cf. *71·17, below), and the hypothesis is equivalent to
These are for many purposes the most convenient hypotheses to use.
The most useful propositions in the present number are the following. (We omit here propositions concerning or which are mere analogues of propositions concerning .)
*71·16.
This gives the connection of one-many relations with descriptive functions. We have also
*71·163.
For many of the constant relations defined from time to time, such as or , the following proposition is useful:
*71·166.
*71·17.
This might have been taken as the definition of one-many relations, if we had not wished to derive them from the more general notion of . In proving that a relation is one-many, *71·17 is more often employed than any other proposition.
*71·22.
*71·25.
*71·36.
*71·381.
(This proposition is more useful than the corresponding property of )
*71·55.
This proposition is constantly used. For example, putting for , it gives
Most of the relations used to establish correlations in arithmetic are obtained from a one-many relation, such as , by imposing some limitation on the converse domain which makes the relation one-one.
*71·571.
Here "" is , which has already played a large part as a hypothesis, e.g. in *37·6 ff.
*71·7.
Thus for example we shall have .
*71·01.
*71·02.
*71·03.
*71·04.
*71·1.
*71·101.
*71·102.