Volume 1 / Part 2 — *53. MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES.
Reading edition
*53. MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES.
Summary of *53.
The propositions to be given in this number are mostly such as would have come more naturally at an earlier stage, but could not be given sooner because they involved unit classes. It is to be observed that is the class consisting of the members and , while is the relation which holds only between and . If and are classes, is a class of classes, its members being and . If and are relations, is a relation of relations; and so on.
The present number begins by connecting products and sums , , , , in cases where the members of or are specified, with the products or sums , , , . We have
*53·01.
*53·1.
*53·14.
with similar propositions for , and .
We have next a set of propositions on sums and products of classes of unit classes. The most important of these is
*53·22.
We have next a proposition showing that the sum of is null when, and only when, is either null or has the null-class for its only member, i.e.
*53·24.
(Here we write "," to show that the "" in question is of the next type above that of the other two 's.)
We have next various propositions on the relations of and and in various cases, first for a general relation , and then for the particular relation defined in *40. Three of these propositions are very frequently used, namely:
*53·3.
*53·301.
*53·31.
The remaining propositions of this number are of less importance, and are seldom referred to.
*53·01.
Dem.
*53·02.
Dem.
*53·03.
*53·04.
*53·1.
Dem.
This proposition can be extended to , etc. It shows the connection (for finite classes of classes) between the product and the product of the members
*53·11.
Dem.
Similar remarks apply to this proposition as to *53·1.
*53·12.
This proposition shows the connection between the product for a class consisting of two relations and , and the product . The proposition can be extended to the product of any given finite class of relations.
*53·13.
Similar remarks apply to this proposition as to *53·12.
*53·14.
Dem.
*53·15.
*53·16.
*53·17.
The above proposition and the next are both used in connection with mathematical induction (*91·55 and *97·46 respectively).
*53·18.
Dem.
*53·181.
*53·2.
This proposition requires, for significance, that should be a class of classes. It is used in *88·47, in the number on the existence of selections and the multiplication axiom.
Dem.
*53·21.
This proposition requires, for significance, that should be a class of relations.
*53·22.
Dem.
*53·221.
Dem.
*53·222.
Dem.
*53·23.
Dem.
*53·231.
Dem.
*53·24.
Dem.
In the enunciation and the last line of the proof of the above proposition, we write "" rather than "," because this must be of the type next above that of the in "."
The following proposition is used in the theory of selections (*83·731).
*53·25.
Dem.
*53·3.
Dem.
The above proposition is very frequently used.
*53·301.
Dem.
*53·302.
The above proposition is used in the cardinal theory of exponentiation (*116·71).
*53·31.
The above proposition is one of which the subsequent use is frequent.
Dem.
*53·32.
Dem.
*53·33.
*53·34.
*53·35.
Dem.
The above proposition may also be proved as follows:
*53·4.
Dem.
*53·5.
Dem.
In the above proof, as usually where "Cls" or other type-symbols occur, it is necessary to abandon the notation by Greek letters and revert to the explicit notation.
*53·51.
*53·52.
Dem.
*53·53.
The following propositions are inserted because of their connection with the definition of in *70. and ( are both important classes.
*53·6.
Dem.
*53·601.
Dem.
*53·602.
*53·603.
*53·604.
*53·61.
Dem.
*53·611.
*53·612.
*53·613.
*53·614.
Dem.
*53·615.
The two following propositions are used in *70·12.
*53·62.
Dem.
*53·621.
*53·63.
*53·631.
*53·64.
*53·641.