*84. CLASSES OF MUTUALLY EXCLUSIVE CLASSES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
*84·41.
This is an important proposition, since it shows that, when is a , the number of classes that can be selected from is the product of the numbers of the various classes that are members of .
*84·411.
*84·412.
This proposition gives what might be taken as the definition of the class of selected classes, namely
We might, starting with this as our definition, deal with the class of selected classes without first considering selective relations. The disadvantages of this method would be, first, that it requires that should be a if it is to give the results desired in arithmetic; secondly, that it is much more cumbrous technically than the method which proceeds by selective relations; thirdly, that it does not enable us to deal with selection from a class of classes as a particular case of selection from a relation (namely from , and therefore does not yield theorems of such generality as those obtained by the method adopted above.
*84·42.
*84·421.
*84·422.
*84·43.
Dem.
*84·5.
Dem.
It might be supposed that the converse of the above would also hold. But this is not the case; for although secures that and cannot overlap when they are unequal, yet we may have without having , so that if , we shall have , whence, if , it follows that is not a even if .
*84·51.
Dem.
*84·52.
Dem.
*84·521.
Dem.
The above proposition is a lemma for *84·522, which is used in an important proposition on relations of mutually exclusive relations (*163·17).
*84·522.
Dem.
*84·53.
Dem.
*84·54.
*84·55.
*84·59.
Dem.
*84·6.
*84·61.
*84·62.