Volume 1 / Part 4 — *165. RELATIONS OF RELATIONS OF COUPLES.
Reading edition
*165. RELATIONS OF RELATIONS OF COUPLES.
Summary of *165.
In the present number, we shall give various propositions concerning the relation , which has the same uses in relation-arithmetic has in cardinal arithmetic. The propositions of this number will be used in the next number to establish the properties of the arithmetical product of two relations and , which is defined as . Again in connection with exponentiation the propositions of the present number will be useful, since, after the product of a relation of relations has been defined (*172), we shall define exponentiation by means of the definition There will also be occasional uses of the propositions of this number throughout the theory of series. The relation is important because its structure is thoroughly known. It is a which consists of relations, each like (*165·27); and if , we can construct a double correlator of and without invoking the multiplicative axiom. In fact we have
*165·362.
This proposition should be compared with *113·127. In virtue of *164·31, together with various propositions of *165 and *166, it will appear that *165·362 includes *113·127 as part of what it asserts.
In the present number, we begin with a set of propositions on fields. We have
*165·12.
*165·13.
whence
*165·14.
which connects the theory of with that of (*113 and *116). Hence
*165·16.
In *166, we shall define as ; thus the above will become
We next have a set of propositions concerned with as a relation, and with the circumstances under which we can infer or from data as to and . We have
*165·21.
*165·211.
*165·22.
We then have various propositions concerning , of which the chief are
*165·241.
*165·242.
We have next four propositions which are constantly used, proving that consists of relations each like . These propositions are
*165·25.
*165·251.
*165·26.
*165·27.
From *165·3 to *165·372, we are concerned with constructing a double correlator of and when we are given simple correlators of with and of with . The result (*165·362) has already been given. Hence we have
*165·37.
and by *164·48 and *165·27 we have
*165·38.
Hence propositions concerning a series of series, each containing terms (where and are relation-numbers), which in general require the multiplicative axiom, can be deduced, assuming that axiom, from propositions (not requiring the axiom) concerning , where and . Thus the use of enables us to minimize the use of the multiplicative axiom.
*165·01.
*165·1.
*165·11.
*165·12.
*165·13.
*165·131.
*165·14.
*165·15.
*165·16.
*165·161.
Dem.
*165·162.
Dem.
*165·17.
Dem.
*165·18.
*165·181.
*165·182.
*165·19.
*165·2.
*165·201.
Dem.
*165·202.
*165·203.
*165·204.
Dem.
*165·205.
*165·206.
*165·21.
Dem.
*165·211.
*165·212.
Dem.
*165·22.
Dem.
*165·221.
Dem.
*165·222.
*165·223.
Dem.
*165·23.
Dem.
*165·231.
*165·232.
Dem.
*165·233.
*165·24.
Dem.
*165·241.
*165·242.
Dem.
*165·243.
Dem.
*165·244.
Dem.
*165·245.
*165·25.
*165·251.
*165·26.
*165·27.
The following propositions are concerned in proving that, if is a correlator of and and is a correlator of and then (with its converse domain limited) is a double correlator of and .
This proposition is required subsequently in establishing likenesses.
*165·3.
Dem.
*165·301.
Dem.
*165·302.
Dem.
*165·31.
Dem.
*165·311.
*165·32.
Dem.
*165·321.
*165·33.
Dem.
*165·331.
*165·34.
Dem.
*165·341.
*165·35.
Dem.
*165·351.
Dem.
*165·352.
Dem.
*165·36.
Dem.
*165·361.
The proof proceeds as in *165·36.
*165·362.
The above three propositions are of great utility in relation-arithmetic.
*165·37.
*165·38.
Dem.