Among one-many relations, one-one relations are a specially important class. We have already had occasion to speak of one-one relations in connection with the definition of number, but it is necessary to be familiar with them, and not merely to know their formal definition. Their formal definition may be derived from that of one-many relations: they may be defined as one-many relations which are also the converses of one-many relations, i.e. as relations which are both one-many and many-one. One-many relations may be defined as relations such that, if has the relation in question to , there is no other term ' which also has the relation to . Or, again, they may be defined as follows: Given two terms and ', the terms to which has the given relation and those to which ' has it have no member in common. Or, again, they may be defined as relations such that the relative product of one of them and its converse implies identity, where the "relative product" of two relations and is that relation which holds between and when there is an intermediate term , such that has the relation to and has the relation to . Thus, for example, if is the relation of father to son, the relative product of and its converse will be the relation which holds between and a man when there is a person , such that is the father of and is the son of . It is obvious that and must be [Pg 47] the same person. If, on the other hand, we take the relation of parent and child, which is not one-many, we can no longer argue that, if is a parent of and is a child of , and must be the same person, because one may be the father of and the other the mother. This illustrates that it is characteristic of one-many relations when the relative product of a relation and its converse implies identity. In the case of one-one relations this happens, and also the relative product of the converse and the relation implies identity. Given a relation , it is convenient, if has the relation to , to think of as being reached from by an "-step" or an "-vector." In the same case will be reached from by a "backward -step." Thus we may state the characteristic of one-many relations with which we have been dealing by saying that an -step followed by a backward -step must bring us back to our starting-point. With other relations, this is by no means the case; for example, if is the relation of child to parent, the relative product of and its converse is the relation "self or brother or sister," and if is the relation of grandchild to grandparent, the relative product of and its converse is "self or brother or sister or first cousin." It will be observed that the relative product of two relations is not in general commutative, i.e. the relative product of and is not in general the same relation as the relative product of and . E.g. the relative product of parent and brother is uncle, but the relative product of brother and parent is parent.