Volume 1 / Part 4 — *150. INTERNAL TRANSFORMATION OF A RELATION.
Reading edition
*150. INTERNAL TRANSFORMATION OF A RELATION.
Summary of *150.
In this number we introduce two notations which have uses in regard to relations closely analogous to the uses of and in regard to classes. These two notations are defined as follows: We then have .
is merely an alternative to , just as is an alternative to . Also , in virtue of *38·01 and *43·01.
The uses of occur chiefly when is a one-one relation and . This case is illustrated in the figure in the introduction to this section. Here if relates and , relates and . Thus given a class similar to , if is the correlating relation, has for its field, and has, in very many respects, properties analogous to those of .
is important for many special values of . For example, let be a relation between relations; then will be the corresponding relation of the fields of these relations. If be any relation, will be the corresponding relation between ordered couples of which is the relatum; i.e. if , the relation will hold between and . If is a relation between classes, and we have , then the relation will hold between and . In short, whenever is a one-many relation, and therefore gives rise to a descriptive function, then is the relation which holds between and whenever holds between and .
We introduce one other new notation in this number, corresponding to in *38. This notation is thus defined: The purpose of this notation is to enable us to proceed to and other similar notations; or, otherwise stated, to enable us to treat a function of y rather than of . Take for example the case of . We may wish to consider various relations , , where we are to have (say) . To express the relation of to resulting from , we need the above notation. By its help, we have Thus is the relation between and corresponding to the relation between and . plays the same part in relation-arithmetic as is played by in cardinal arithmetic.
The notations of this number are capable of occasional uses in cardinal arithmetic[11], but their chief utility is in relation-arithmetic, in which they are fundamental.
In order to minimize the use of brackets, we put
As an immediate result of the definition of we have
*150·11.
We have also
*150·12.
*150·13.
This proposition, which is the analogue of (*37·33), is very often used. We have also
*150·3.
*150·42.
The remaining propositions of this number (with a few exceptions) may be thus classified:
(1) Propositions concerning the domain, converse domain, and field of (*150·2—·23). Owing to the fact that the chief applications of this subject are to cases where and are serial, the field of is more important than its domain or converse domain. Thus the chief propositions here are
*150·22.
*150·23.
The hypothesis is verified in almost all applications of . When it is not verified, the part of not contained in is irrelevant to the value of . The hypothesis is very often verified in practice, since it is verified when is a correlator of and .
(2) Propositions concerning relations with limited domains, converse domains, or fields (*150·32—·38). Broadly speaking, a limitation on the field of is equivalent to a limitation on the converse domain of , and both are equivalent to a corresponding limitation on the field of provided . The limitations that occur in practice are limitations on the converse domain of , with consequent limitations on the fields of and .
The chief propositions on this subject are
*150·32.
*150·35.
(This follows from *150·32 and *35·71.)
*150·36.
*150·37.
(3) Propositions on when is one-many or many-one (*150·4—·56). We have
*150·4.
This proposition is used constantly. Only slightly less useful is
*150·41.
The remaining propositions of this set are chiefly applications of *150·4·41 to special cases.
(4) A few propositions on (*150·6—·62). These are immediate consequences of the definition.
(5) A set of propositions on couples and matters connected with them (*150·7—·75). The chief of these is
*150·71.
This proposition is very often used in relation-arithmetic. Useful also is
*150·73.
(6) We next have four propositions (*150·8—·83) on when is a power of . These belong with the propositions of *92; they are useful in the ordinal theory of finite and infinite. We have
*150·82·83.
It follows that, in the hypothesis supposed, if is a correlator of and , it is also a correlator of and .
(7) Propositions concerning the relation (*150·14—·171 and *150·9—·94). These have uses analogous to those of propositions concerning . The most important are
*150·14.
(This follows immediately from *150·13, above.)
*150·141.
(This follows immediately from the definition.)
*150·16.
This proposition is analogous to (*40·38), i.e. to as appears on substituting and for and in this variant of *40·38.
The remaining propositions are mainly of the nature of lemmas, to be used once or twice each in relation-arithmetic.
*150·01.
*150·02.
*150·03.
Here, as in *38, "" stands for any sign which, when placed between two letters, defines a descriptive function of the arguments represented by those letters. Thus for example "" may represent any of the following:
The two following definitions serve merely for the avoidance of brackets.
*150·04.
*150·05.
*150·1.
*150·11.
*150·12.
*150·13.
Dem.
*150·131.
Dem.
Observe that we do not have .
*150·14.
Dem.
This proposition is the relational analogue of *37·34.
*150·141.
*150·15.
*150·151.
The following proposition is used in the theory of double ordinal similarity (*164·13).
*150·152.
Dem.
*150·153.
Dem.
The above proposition is used in dealing with relations of relations of couples (*165·23).
*150·16.
The following proposition is a lemma for *150·171.
*150·17.
Dem.
*150·171.
Dem.
The above proposition is required in the theory of double ordinal similarity. It is used in proving *164·141, which is used in *164·18, which is a fundamental proposition in the theory of double ordinal similarity.
The following propositions, on the domain, converse domain and field of , are much used, especially *150·202·22·23. *150·201 is hardly ever used, but is inserted in order that the general case may not remain unconsidered.
*150·2.
*150·201.
Dem.
*150·202.
Dem.
*150·203.
*150·21.
*150·211.
*150·22.
In practice, when is used, we almost always have . For the use of is to obtain a relation analogous to and having a different field; now is analogous to , for the part of which lies outside is unaffected by . Hence if we have, to start with, a relation whose field is not contained in , we shall usually find it profitable to limit the field to , and consider the transformed relation rather as than as . Thus the hypothesis will be verified in almost all useful applications of the notion of .
*150·23.
*150·24.
Dem.
*150·25
*150·3.
*150·301.
*150·31.
The following propositions are frequently useful when we have to deal with correlators of the form , which often happens.
*150·32.
*150·33.
*150·34.
*150·35.
Dem.
The above proposition, which is the analogue of *37·69, is much used in relation-arithmetic.
The following proposition is much used after we reach the theory of well-ordered series, but not before (except in *150·37).
*150·36.
Dem.
*150·361.
*150·37.
Dem.
The above proposition is not used until we reach the theory of series.
*150·38.
Dem.
The above proposition is used in dealing with the correlation of series (*208·2).
*150·4.
This proposition is fundamental in the theory of , because in most of the uses of this notion is one-many. The proposition states that when is one-many, is the relation between the of terms related by . Thus if is the relation of wife to husband, and is the relation of brother to brother, is the relation between wives of brothers. If is a relation between relations, will be the corresponding relation of their fields; and so on.
*150·41.
*150·42.
The following propositions, down to *150·56, are, with the exception of *150·52—·535, all illustrations of *150·4·41.
*150·5.
*150·51.
*150·511.
*150·512.
*150·52.
is a relation which plays a great part in relation-arithmetic.
*150·53.
*150·531.
*150·532.
*150·534.
*150·535.
*150·54.
*150·541.
*150·55.
*150·56.
*150·6.
*150·601.
*150·61.
*150·62.
Relations of the form are frequently useful in relation-arithmetic, especially in the particular case of , which takes the place taken by in cardinal arithmetic. Relations of the form will be considered in *165.
The following propositions are chiefly concerned with correlations of couples. They are of great utility in relation-arithmetic. *150·71, in particular, is fundamental.
*150·7.
*150·71.
*150·72.
*150·73.
*150·74.
*150·75.
Dem.
The four following propositions belong to the subject of *92, but could not be given in that number owing to the fact that they involve the notations of *150. They are required for proving that, if is a correlator of and , it is also a correlator of and (*151·45), and for one of the fundamental propositions in the ordinal theory of progressions (*263·17).
*150·8.
Dem.
*150·81.
Dem.
*150·82.
Dem.
*150·83.
Dem.
The following propositions, down to *150·94 inclusive, resume the subject of the relation , which has already been treated in *150·14—·171.
*150·9.
Dem.
The following propositions lead up to *150·931·94, which are used in the theory of double ordinal similarity (*164·3·21).
*150·91.
Dem.
*150·92.
Dem.
*150·921.
*150·93.
*150·931.
Dem.
*150·932.
*150·933.
*150·94.
Dem.
The above proposition is the analogue of *74·61, which (with a few trivial transformations) may be written
In obtaining ordinal analogues of such propositions, will be replaced by , and the two inverted commas will be replaced by the semi-colon; a class of classes will be replaced, in most of its occurrences, by a relation of relations , but will sometimes be replaced by .
The above proposition (*150·94) is used in proving that the converse of a double correlator of and is a double correlator of and (*164·21). The corresponding cardinal proposition (*111·131) uses *74·6, which is practically the same proposition as *74·61, which is the analogue of *150·94.
*150·95.
Dem.
The above proposition is used in the theory of "first differences" (*170·41).
*150·96.
Dem.
*150·961.
Dem.
The above proposition is used in the theory of ordinal exponentiation (*176·21).