Volume 1 / Part 4 — *166. THE PRODUCT OF TWO RELATIONS.
Reading edition
*166. THE PRODUCT OF TWO RELATIONS.
Summary of *166.
The product is defined as . This is a relation which has for its field all the couples that can be formed by choosing the referent in and the relatum in . These couples are arranged by on the following principle: If the relatum of the one couple has the relation to the relatum of the other, we put the one before the other, and if the relata of the two couples are equal while the referent of the one has the relation to the referent of the other, we put the one before the other. Thus in advancing from any term in the field of , we first keep fixed and alter into later terms as long as possible; then we alter into a later term, move back to the beginning, and so on. Thus with a given , we get a series which is like , and this series is wholly followed or wholly preceded by the series with the referent , where follows or precedes .
The propositions of this number are for the most part immediate consequences of those of *165. The most important of them are:
*166·12.
*166·13.
Hence it follows that an ordinal product of a finite number of factors vanishes when, and only when, one of its factors vanishes.
*166·16.
*166·23.
This proposition shows that the relation-number of a product depends only upon the relation-numbers of its factors.
*166·24.
This proposition connects addition and multiplication (cf. note to *166·24, below).
*166·42.
This is the associative law. The distributive law has two forms:
*166·44.
*166·45.
We do not have in general (cf. note before *166·44, below)
We have also a distributive law for the addition of a single term, i.e.
*166·53.
*166·531.
Here again the law does not hold in general for or .
*166·01.
*166·1.
*166·11.
*166·111.
*166·112.
*166·113.
*166·12.
*166·13.
*166·14.
*166·15.
*166·16.
Dem.
The above proposition is used in the ordinal theory of progressions (*263·62·65).
*166·2.
*166·21.
*166·22.
This proposition gives the correlator for the product when correlators are given for the factors.
*166·23.
This proposition enables us to use to define the product of the relation-numbers of and , for it shows that the relation-number of is determinate when the relation-numbers of and are given. We shall therefore (in Section D of this part) define the product of two relation-numbers and as the relation-number of when and .
*166·24.
This proposition exhibits the connection of addition and multiplication. If we put and , then in the above proposition is the sum of relations of which each is a . In virtue of the above proposition, it follows that (if the multiplicative axiom is assumed) . In other words, assuming the multiplicative axiom, the sum of series (or other relations), each of which has terms, has terms.
*166·3.
The analogous proposition is only true in general if
*166·31.
*166·311.
*166·312.
The following propositions are lemmas for the associative law (*166·42).
*166·4.
Dem.
*166·401.
*166·41.
Dem.
*166·42.
This is the associative law for the kind of multiplication concerned in this number.
*166·421.
This definition serves merely for the avoidance of brackets.
The two following propositions give the distributive law. In relation-arithmetic, this is in general only true in one of its two forms, i.e. we have The latter is true for finite series, but not for infinite series or (except in exceptional cases) for relations which are not serial.
*166·44.
Dem.
*166·45.
Dem.
The following propositions (*166·46—·472) exhibit the failure of the distributive law in the form , and give certain results for special cases. They are not referred to except in this number.
*166·46.
*166·461.
*166·462.
Dem.
*166·463.
*166·464.
Dem.
*166·47.
*166·471.
*166·472.
Dem.
The following propositions are concerned with the distributive law for the addition of a single term to a relation. This law, in the form in which it holds, is given in *166·53·531 (remembering ). *166·54·541 exhibit the failure of the other form.
*166·5.
Dem.
*166·51.
*166·511.
*166·52.
*166·521.
*166·53.
Dem.
*166·531.
*166·54.
Dem.
*166·541.