*81. SELECTIONS FROM MANY-ONE RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *81.
When is a many-one relation, has many important properties which do not hold in the general case. In the first place, consists wholly of one-one relations. In the second place, if , takes one term and no more out of each member of . Again, if , is determinate when is given; i.e. . It follows that is similar to ; hence the number of members of is the number of ways of choosing one member out of each class belonging to . It should be remembered that when is many-one, is a class of mutually exclusive classes, i.e. no two different members of have any common member. This follows immediately from *71·181.
As explained in the introduction to this section, the propositions of this number are chiefly useful on account of their application to the case of . This application is made in *84. The most important propositions in this number are:
*81·1.
*81·14.
This proposition, by exhibiting as a function of , leads immediately to
*81·21.
This is the principal proposition of this number. The following also is important:
*81·22.
*81·1.
Dem.
*81·11.
Dem.
*81·12.
Dem.
*81·13.
Dem.
*81·14.
This proposition, by exhibiting as a function of , shows that a member of is determinate when its domain is given, provided .
*81·15.
Dem.
*81·2.
Dem.
*81·21.
This proposition is very important. The class , when , is formed, as we shall prove later, by making every possible selection of one term out of each member of , each such selection giving us one member of . The fact that, with the above hypothesis, the class of classes has the same number of terms as (which results from the above proposition), is of great utility in the theory of cardinal multiplication and exponentiation.
*81·211.
Dem.
*81·212.
Dem.
*81·22.
*81·221.
Dem.
*81·23.
Dem.
*81·24.
Dem.
*81·25.
Dem.
*81·26.
Dem.