Volume 1 / Part 2 — SECTION C. ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS.

Summary of Section C.

In the present section we have to consider three very important classes of relations, of which the use in arithmetic is constant. A one-many relation is a relation such that, if is any member of , there is one, and only one, term which has the relation to , i.e. . Thus the relation of father to son is one-many, because every son has one father and no more. The relation of husband to wife is one-many except in countries which practise polyandry. (It is one-many in monogamous as well as in polygamous countries, because, according to the definition, nothing is fixed as to the number of relata for a given referent, and there may be only one relatum for each given referent without the relation ceasing to be one-many according to the definition.) The relation in algebra of to is one-many, but that of to is not, because there are two different values of that give the same value of .

When a relation is one-many, exists whenever , and vice versa; i.e. we have

Thus relations which give descriptive functions that are existent whenever their arguments belong to the converse domains of the relations in question are one-many relations. Hence , , , , , , , , , , , , , , , , , are all of them one-many relations.

When is a one-many relation, is a one-valued function; conversely, every one-valued function is derivable from a one-many relation. A many-valued function of is a member of , where is not a unit class, and any one of its members is regarded as a value of the function for the argument y; but a one-valued function of is the single term which is obtained when is one-many. Thus for example the sine would, in our notation, appear as a relation, i.e. we should put so that "" has the usual meaning of . Then instead of , we should have , which would be the class of values of ; and instead of "," which is a misleading notation because and do not imply , we should have . Similar remarks would apply to any of the other functions that occur in analysis.

A relation is called many-one when, if is any member of , there is one, and only one, term to which has the relation , i.e. . Thus many-one relations are the converses of one-many relations. When a relation is many-one, exists whenever .

A relation is called one-one when it is both one-many and many-one, or, what comes to the same, when both it and its converse are one-many. Of the one-many relations above enumerated, , , , , , , , are one-one.

Two classes , are said to be similar when there is a one-one relation such that , i.e. when their terms can be connected one to one, so that no term of either is omitted or repeated. We write "" for " is similar to ." When two classes are similar, the cardinal numbers of their terms are the same; it is this fact chiefly that makes one-one relations of fundamental importance in cardinal arithmetic.

According to the above, a relation is one-many when

Similarly a relation is many-one when and a relation is one-one when both conditions are fulfilled. The classes , , which appear here, are often important; some of their properties have already been given in *37·77·771·772·773 and in *53·61 to *53·641.

It is convenient to regard one-many, many-one and one-one relations as particular cases of relations which, for some given and , have

Hence, without a new definition, "" becomes the class of one-one relations; also, as will be shown, "" becomes the class of one-many relations, and "" becomes the class of many-one relations. Although it is chiefly these three special values of that are important, we shall begin by a general study of classes of relations of the form .