Volume 1 / Part 1 — *35. RELATIONS WITH LIMITED DOMAINS AND CONVERSE DOMAINS.
Reading edition
*35. RELATIONS WITH LIMITED DOMAINS AND CONVERSE DOMAINS.
Summary of *35.
In this section, we have to consider the relation derived from a given relation by limiting either its domain or its converse domain to members of some assigned class. A relation with its domain limited to members of is written ""; with its converse domain limited to members of , it is written ""; with both limitations, it is written "." Thus e.g. "brother" and "sister" express the same relation (that of a common parentage), with the domain limited in the first case to males, in the second to females. "The relation of white employers to coloured employees" is a relation limited both as to its domain and as to its converse domain. We put
*35·01.
with similar definitions for and .
A particularly important case is the case in which the same limitation is imposed on the domain and on the converse domain, i.e. where we have a relation of the form "." In this case, the limitation to members of may be more briefly stated as being imposed on the field. For this case, it is convenient to adopt "" as an alternative notation. This case will be considered in *36.
It is convenient to consider in the present connection the relation between and which is constituted by x being a member of and being a member of . This relation will be denoted by "." Thus we put
*35·04.
The chief importance of relations with limited fields arises in the theory of series. Given a series generated by a relation , let be a class consisting of part of this series. Then is the field of the relation or , and it is this relation which is the generating relation of the series of members of in the same order which they have as parts of the original series. Thus parts of a series, considered not merely as classes but as series, are dealt with by means of serial relations with limited fields.
Relations with limited domains are not nearly so much used as relations with limited converse domains. Relations with limited converse domains play a great part in arithmetic, especially in establishing the formal laws. What is wanted in such cases is a one-one relation correlating two classes or two series. That is, we want a relation such that not only does exist whenever , but also exists whenever . The kind of relation which is most frequently found to effect such a correlation is some such relation as or or , or some other constant relation for which we always have , with its converse domain so limited that, subject to the limitation, only one value of gives any given value of . Thus for example let be a class of relations no two of which have the same domain; then will give a one-one correlation of these relations with their domains: if , , we shall have We shall also have and . Moreover the converse domain of is , and the domain of is the class of domains of members of . Thus gives a one-one correlation of with the domains of members of . It is chiefly in such ways that relations with limited converse domains are useful.
For purposes of reference, a great many propositions are given in the present number, but the propositions that will be used frequently are comparatively few. Among these are the following:
*35·21.
*35·31.
*35·354.
I.e. in a relative product it makes no difference whether we limit the converse domain of the first factor, or the domain of the second.
*35·412.
*35·452.
*35·48.
*35·52.
*35·61.
*35·64.
*35·65.
The hypothesis is fulfilled in the great majority of cases in which we have occasion to use .
*35·66.
*35·7.
This proposition is used very frequently, owing to the fact that limitation of the converse domain is chiefly applied to such relations as give rise to descriptive functions (e.g. , , ).
*35·71.
This proposition is useful for a reason similar to that which makes *35·7 useful.
*35·82.
Owing to this proposition, the properties of can be deduced from the already proved properties of , by putting .
The relation "" is what may be called an "analysable" relation, i.e. it holds between and y when and , i.e. when has a property independent of , and has a property independent of .
*35·85.
*35·86.
If either or is null, so is (*35·88).
*35·01.
*35·02.
*35·03.
*35·04.
*35·05.
The last definition serves merely for the avoidance of brackets.
*35·1.
*35·101.
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Dem.
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Dem.
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Dem.
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Dem.
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*35·18.
*35·21.
Dem.
*35·22.
Dem.
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*35·25.
*35·26.
Dem.
*35·27.
*35·31.
Dem.
*35·32.
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*35·34.
*35·35.
Dem.
*35·351.
*35·352.
*35·354.
Dem.
*35·41.
*35·412.
*35·413.
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*35·421.
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*35·43.
Dem.
*35·431.
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*35·44.
Dem.
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*35·451.
Dem.
*35·452.
*35·453.
*35·454.
*35·46.
Dem.
*35·461.
*35·462.
*35·471.
Dem.
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*35·473.
*35·474.
*35·48.
Dem.
*35·481.
*35·51.
Dem.
*35·52.
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*35·61.
Dem.
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Dem.
*35·64.
*35·641.
*35·642.
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*35·644.
*35·65.
*35·66.
*35·671.
Dem.
*35·672.
*35·68.
Dem.
*35·7.
This proposition is very often used in the later parts of the work.
Dem.
*35·71.
Dem.
*35·75.
Dem.
*35·76.
Dem.
The rest of this number, down to *35·93 exclusive, is concerned with , except *35·81·812.
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*35·812.
*35·82.
Dem.
*35·822.
Dem.
*35·83.
Dem.
*35·831.
Dem.
*35·832.
*35·834.
Dem.
*35·84.
*35·85.
Dem.
*35·86.
*35·87.
Dem.
*35·88.
*35·881.
Dem.
*35·882.
*35·89.
Dem.
*35·891.
Dem.
*35·892.
*35·895.
*35·9.
Dem.
*35·91.
Dem.
*35·92.
*35·93.
Dem.
*35·931.
*35·932.
*35·94.
*35·941.
*35·942.