Volume 1 / Part 2 — *84. CLASSES OF MUTUALLY EXCLUSIVE CLASSES.
Reading edition
*84. CLASSES OF MUTUALLY EXCLUSIVE CLASSES.
Summary of *84.
A class of mutually exclusive classes is one such that, if and are two different members of , and have no common members; i.e. it is a class composed of non-overlapping classes. Classes of mutually exclusive classes have many important properties. They are important in cardinal arithmetic, among other reasons, because if is a class of mutually exclusive classes, the cardinal number of is the sum of the cardinal numbers of the members of . Also if is a class of mutually exclusive classes, the number of selected classes of (i.e. ) is the same as the number of selective relations (i.e. )
" is a class of mutually exclusive classes" is written "."
An important case is when no member of is null; in this case we write
For a which is contained in a class of classes , we write on the analogy of the notation .
The definitions are as follows:
*84·01.
*84·02.
*84·03.
The propositions of this number begin (*84·1—·14) with various equivalent forms for the definitions. Of these the most useful are:
*84·11.
*84·13.
*84·14.
The last of these is specially important, because it renders the propositions of *81 applicable to when .
We have next (*84·2—·28) a set of propositions dealing with various special cases, such as and 1. The most useful of these are
*84·23.
*84·241.
*84·25.
We next have a set of propositions (*84·3—·37) which are immediate consequences of propositions in *81, by means of *84·14. The most useful of these is
*84·3.
We next have a set of propositions (*84·4—·43) dealing with the domains of selections from a . These are for the most part still immediate consequences of propositions in *81, in virtue of *84·14. The most useful are
*84·41.
*84·412.
*84·43.
This proposition applies to such cases as the relations of rows and columns. Imagine any set of terms arranged in rows and columns so as to form a rectangle. Then each column is a selection from the rows, and each row is a selection from the columns. This is a particular case of the above proposition.
We next have a set of propositions on , , and (*84·5—·55). The most important of these are
*84·51.
*84·53.
Finally we have a set of propositions (*84·59—·62) showing circumstances under which is a . The only one of these which is used subsequently is
*84·62.
*84·01.
*84·02.
*84·03.
*84·1.
*84·11.
*84·12.
*84·121.
*84·13.
Dem.
*84·131.
*84·132.
*84·133.
*84·134.
Dem.
*84·135.
Dem.
*84·14.
Dem.
This proposition is important, since it enables us to apply the propositions of *81 to when .
*84·2.
Dem.
*84·21.
Note. is the class of all unit classes whose members are classes; this results from *65·02. Thus "" is equivalent to " consists of one class."
Dem.
*84·22.
Dem.
*84·23.
*84·24.
Dem.
*84·241.
Dem.
*84·242.
*84·25.
Dem.
*84·26.
Dem.
*84·28.
Dem.
The following propositions are concerned with selections from a . In virtue of *84·14, the propositions of *81 which have the hypothesis become applicable when is and is a . Thus has many important properties when is a which it does not have in the general case.
*84·3.
*84·31.
*84·32.
*84·33.
*84·34.
*84·341.
*84·342.
*84·35.
Dem.
*84·37.
*84·4.
*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.