Volume 1 / Part 1 — *41. THE PRODUCT AND SUM OF A CLASS OF RELATIONS.
Reading edition
*41. THE PRODUCT AND SUM OF A CLASS OF RELATIONS.
Summary of *41.
The propositions to be given in this number, down to *41·3 exclusive, are the analogues of those of *40, excluding those from *40·3 onwards, which have no analogues. Proofs will not be given, in this number, when they are exactly analogous to those of propositions with the same decimal part in *40. The smaller importance of and , as compared with and , is illustrated by the smaller number of propositions in *41 as compared with *40.
Our definitions are
*41·01.
*41·02.
Of the propositions preceding *41·3, which are analogues of propositions in *40, the only two that are frequently used are
*41·13.
*41·151.
Of the remaining propositions of this number, which have no analogues in *40, the most important are *41·43·44·45, namely These propositions are constantly required in the theory of selections (Part II, Section D) and in relation-arithmetic. Most of the other propositions of this number are used only once or not at all.
*41·01.
*41.02.
*41·1.
*41·11.
*41·12.
*41·13.
*41·14.
*41·141.
*41·15.
*41·151.
*41·16.
*41·161.
*41·17.
*41·171.
*41·18.
*41·181.
*41·19.
*41·2.
*41·21.
*41·22.
*41·221.
*41·23.
*41·24.
*41·25.
*41·26.
*41·27.
*41·3.
Dem.
*41·31.
*41·32.
*41·33.
*41·34.
Dem.
*41·341.
*41·342.
Dem.
The following proposition is used in *85·22.
*41·35.
Dem.
*41·351.
*41·4.
Dem.
*41·41.
*41·42.
Dem.
*41·43.
Dem.
*41·44.
*41·45.
Dem.
*41·5.
Dem.
*41·51.
Dem.
The above proposition, which is used in *92·31, states that, if and are classes of relations, the relative product of the relational sum of and the relational sum of is the relational sum of all the relative products formed of a member of and a member of .
The following proposition is used in *96·111.
*41·52.
Dem.
The following proposition is used in *162·32 and in *166·461.
*41·6.
Dem.