Volume 1 / Part 4 — *160. THE SUM OF TWO RELATIONS.
Reading edition
*160. THE SUM OF TWO RELATIONS.
Summary of *160.
In this number, we introduce the definition which was explained in the introduction to this section. Although the propositions of this and other numbers in this Part do not require that and should be such as to generate series, yet the reader will find it convenient to imagine them to be such, since the important applications of the ideas of this Part are to series. Thus we may regard the sum of and as a relation which holds between and when either precedes in the -series, or precedes in the -series, or belongs to the -series and belongs to the -series.
The most important propositions of this number are :
*160·14.
*160·21.
*160·22.
*160·31.
which is the associative law, and
*160·4.
which is the distributive law for logical and arithmetical addition;
*160·44.
which is also a kind of distributive law;
*160·47.
whence
*160·48.
whence it follows that if and are mutually exclusive, the relation-number of their sum depends only upon the relation-numbers of and ;
*160·5.
*160·52.
*160·01.
*160·1.
*160·11.
*160·111.
*160·12.
*160·13.
*160·14.
Dem.
The above proposition is constantly used. The following propositions (*160·15—·161) are not used, but are inserted to show that has the kind of structure that we should expect of a sum.
*160·15.
Dem.
*160·151.
*160·16.
*160·161.
*160·2.
*160·21.
*160·22.
*160·3.
Dem.
*160·31.
Dem.
*160·32.
This definition serves merely for the avoidance of brackets.
*160·33.
*160*34.
*160·35.
*160·4.
Dem.
*160·401.
The above two propositions state the distributive law for logical and arithmetical addition. The three following propositions give the generalized form of this law, when replaces ; these propositions are not subsequently used but are inserted for the sake of their intrinsic interest.
*160·41.
Dem.
*160·411.
*160·412.
Dem.
The following propositions lead up to *160·44, which is frequently used.
*160·42.
Dem.
*160·421.
*160·43.
Dem.
*160·44.
Dem.
*160·45.
Dem.
*160·451.
Dem.
*160·452.
Dem.
*160·46.
Dem.
*160·47.
Dem.
*160·48.
*160·5.
Dem.
*160·51.
Dem.
The above proposition is useful in proving that, if , is transitive when and are transitive (cf. *201·4).
*160·52.
Dem.
The above proposition is used in dealing with the series of segments of a series (*213·561).