A "selector" from a class of classes is a one-many relation, having for its converse domain, and such that, if has the relation to , then is a member of . If is a selector from , and is a member of , and is the term which has the relation to , we call the "representative" of in respect of the relation . A "selection" from will now be defined as the domain of a selector; and the multiplicative class, as before, will be the class of selections. But when the members of overlap, there may be more selectors than selections, since a term which belongs to two classes and may be selected once to represent and once to represent , giving rise to different selectors in the two cases, but to the same selection. For purposes of defining multiplication, it is the selectors we require rather than the selections. Thus we define: "The product of the numbers of the members of a class of classes " is the number of selectors from . We can define exponentiation by an adaptation of the above [Pg 120] plan. We might, of course, define as the number of selectors from classes, each of which has terms. But there are objections to this definition, derived from the fact that the multiplicative axiom (of which we shall speak shortly) is unnecessarily involved if it is adopted. We adopt instead the following construction:— Let be a class having terms, and a class having terms. Let be a member of , and form the class of all ordered couples that have for their second term and a member of for their first term. There will be such couples for a given , since any member of may be chosen for the first term, and has members. If we now form all the classes of this sort that result from varying , we obtain altogether classes, since may be any member of , and has members. These classes are each of them a class of couples, namely, all the couples that can be formed of a variable member of and a fixed member of . We define as the number of selectors from the class consisting of these classes. Or we may equally well define as the number of selections, for, since our classes of couples are mutually exclusive, the number of selectors is the same as the number of selections. A selection from our class of classes will be a set of ordered couples, of which there will be exactly one having any given member of for its second term, and the first term may be any member of . Thus is defined by the selectors from a certain set of classes each having terms, but the set is one having a certain structure and a more manageable composition than is the case in general. The relevance of this to the multiplicative axiom will appear shortly.