Volume 1 / Part 1 — *40. PRODUCTS AND SUMS OF CLASSES OF CLASSES.

*40. PRODUCTS AND SUMS OF CLASSES OF CLASSES.

Summary of *40.

In this number, we introduce the two notations (explained above) Both these notions will be found increasingly useful as we proceed, but remains more useful than throughout. It is required for the significance of and that should be a class of classes.

In the present number, the most useful propositions are the following:

*40·12.

I.e. the product of is contained in every member of .

*40·13.

I.e. every member of is contained in the sum of .

*40·15.

I.e. is contained in the product of if is contained in every member of , and vice versa.

*40·151.

I.e. the sum of is contained in if every member of is contained in , and vice versa.

*40·2.

I.e. the product of the null-class of classes is the universal class. This may seem paradoxical at first sight, but it is really not so. The fewer members has, the larger, speaking generally, becomes. If has no members, then has no members to which a given term does not belong, and therefore belongs to .

*40·23.

I.e. unless is null, its product is contained in its sum.

*40·38.

This proposition is very often used in arithmetic. What it states is as follows: Given a class of classes , take its sum, , and then consider all the terms that have the relation to some member of ; this gives the class ; next, take each separate member of , say , and form the class , consisting of all terms having the relation to some member of . The class of all such classes as , for various 's which are members of , is ; the sum of this class, by the above proposition, is the same as .

*40·4.

This proposition requires, for significance, that should always be a class. The proposition states that, if always exists when , then the sum of all classes which have the relation to some member of consists of all members of such classes as , where .

*40·5.

This proposition results from *40·4 by substituting for in that proposition.

*40·51.

In virtue of *40·5, is correlative to . Thus if is a serial relation, consists of terms preceding the whole of , and consists of terms preceding part of . If has a lower limit, it will be the upper limit or maximum of ; if has an upper limit, it will be the upper limit of .

*40·61.

In this proposition the hypothesis is essential, since, if , and .

*40·01.

*40·02.

*40·1.

*40·11.

*40·12.

Dem.

*40·13.

Dem.

*40·14.

*40·141.

*40·15.

Dem.

*40·151.

Dem.

This proposition is frequently used.

*40·16.

Dem.

*40·161.

Dem.

*40·17.

Dem.

*40·171.

Dem.

*40·18.

Dem.

*40·181.

Dem.

*40·19.

This proposition is the extension of *22·6.

Dem.

*40·2.

Dem.

*40·21.

Dem.

In the above proposition, the two 's are of different types, since is of the type next above that of . Thus it would be more correct to write

But in the case of it is not very important to keep the types distinct.

*40·22.

Dem.

In this proposition, the two 's are of the same type.

*40·221.

Dem.

Observe that the hypothesis is essential to this proposition, since when , and . Thus

*40·23.

Dem.

Observe that the hypothesis is essential to this proposition, since when , and . Thus

*40·24.

Dem.

The above proposition is used in the proof of *215·25.

*40·25.

Dem.

*40·26.

Dem.

The following proposition is used in the proof of *216·51.

*40·27.

Dem.

The following propositions are only significant when is a relation whose domain consists of classes, for they concern or , and therefore require that should be a class of classes.

*40·3.

*40·31.

*40·32.

Dem.

*40·33.

The following propositions no longer require that the domain of should be composed of classes.

*40·35.

Dem.

*40·36.

*40·37.

Dem.

*40·38.

Dem.

This proposition is frequently used in the proofs of arithmetical propositions.

*40·4.

This proposition is only significant when .

Dem.

*40·41.

*40·42.

Dem.

This proposition is used in *40·57, where we take , , .

*40·43.

Dem.

*40·44.

Dem.

The following proposition is used in the proof of *84·44.

*40·45.

Dem.

The following proposition is used in the proof of *94·402.

*40·451.

Dem.

*40·5.

Dem.

*40·51.

is the class of terms each of which has the relation to every member of , just as is the class of terms each of which has the relation to some member of . In the theory of series, plays an important part, correlative to that played by (which is , by *40·5). If is a class contained in a series whose generating relation is , then will be the predecessors of all members of , while will be the predecessors of some .

*40·52.

*40·53.

*40·54.

*40·55.

From this point onwards to *40·69, the propositions are inserted on account of their use in the theory of series.

*40·56.

In the above proposition, the conditions of significance require that should be a class of relations.

*40·57.

*40·6.

*40·61.

Dem.

*40·62.

The two following propositions (*40·63·64) are used in proving *40·65, which is used in *204·63.

*40·63.

Dem.

*40·64.

*40·65.

*40·66.

Dem.

*40·67.

*40·68.

Dem.

This proposition is used in the theory of series (*206·2).

*40·681.

The following proposition is used in *211·56.

*40·682.

Dem.

*40·69.

Dem.

The above propositions concerning and of course have analogues for and . But owing to *40·5, these analogues are more simply stated as properties of and . Thus, for example, *37·264 is the analogue of *40·67. The above propositions concerning and will be used in the theory of series, but until we reach that stage they will seldom be referred to.

*40·7.

Dem.

This proposition is of considerable importance, since it gives a compact form for the class of all values of the function obtained by taking in the class and in the class. Thus, for example, suppose is the class of numbers which are multiples of 3, and is the class of numbers which are multiples of 5, and represents the arithmetical product of and , then will be the class of products of multiples of 3 and multiples of 5, i.e. the class of multiples of 15. Again suppose and are both classes of relations; then will be all relative products obtained by choosing in the class and in the class .

*40·71.

Dem.

The hypothesis , which appears in *40·8·81, is one which plays an important part at a later stage. In the theory of induction (Part II, Section E) it characterizes a hereditary class, and in the theory of series it characterizes an upper section (when combined with ).

*40·8.

Dem.

*40·81.

Dem.