Volume 1 / Part 4 — *163. RELATIONS OF MUTUALLY EXCLUSIVE RELATIONS.
Reading edition
*163. RELATIONS OF MUTUALLY EXCLUSIVE RELATIONS.
Summary of *163.
In the present number we have to define mutually exclusive relations, and to give a few of their properties. Mutually exclusive relations play much the same part in relation-arithmetic as mutually exclusive classes play in cardinal arithmetic. Prima facie, there are various ways in which we might define them. We might define as a relation of mutually exclusive relations when or when or in several other ways. But in fact the most useful property to choose is the property that any two members of the field have mutually exclusive fields, i.e.
The principal applications of the subjects studied in this Part are to series, and in series it is always the fields of the relations that are important. We want, for instance, to define relations of mutually exclusive relations in such a way that, if is a serial relation, and every member of is a serial relation, then is a serial relation. For this purpose it is necessary that should be contained in diversity, which requires that should be contained in diversity, i.e. that If is a serial relation, as we are supposing, this is equivalent to
Again we want to define relations of mutually exclusive relations in such a way that, if and are two such relations, and and have double likeness (cf. *164), then is like ; i.e. if we are given a correlator of and , and for every and which correlates, we are again given a correlator, then is to be like . That is, if is the class of relations which correlate pairs of relations and , where , we want to be a correlator of and . Now this requires that should be a one-one relation, which requires This is secured by but except for special classes of relations it is not secured by since there may be two relations and which both belong to the field of , but of which neither has the relation to the other. Again, the analogy with cardinal arithmetic fails at many points unless, when is a relation of mutually exclusive relations, is a class of mutually exclusive classes. But this is not secured by any of the other possible definitions we have been considering. There are further reasons, connected with the arithmetical product of a relation of relations, for choosing as the definition
From a technical point of view, the properties of a depend mainly upon the fact that when is such a class, (*84·14); in like manner the properties of a depend upon which requires our definition, and is equivalent to it (*163·12). We thus become able to use the propositions of *81 on selections from many-one relations, which would not otherwise be the case.
It should be observed that is not equivalent to though it implies this. The converse implication will fail if contains two different relations with the same field. E.g. take a relation whose field consists of the four relations , , , , and suppose . Then , and . But unless and we shall not have
The property by which we define relations of mutually exclusive relations is a property which only depends on the field, so that we might equally well put But for our purposes this would be less convenient than the definition of .
We thus put
*163·01.
We have
*163·11.
*163·12.
*163·17.
Any of the above might have been used to define . The following propositions are important.
*163·3.
This is the analogue of *84·53.
*163·4·41.
*163·441.
*163·451.
*163·01.
*163·1.
*163·11.
*163·12.
For many purposes, this proposition gives the most useful equivalent of .
Instead of the above proof, we may use *74·62, which gives us the result in virtue of *33·5.
*163·13.
*163·14.
*163·15.
Dem.
*163·16.
*163·17.
*163·2.
*163·21.
Dem.
This proposition is important in connection with the multiplication of relations, for we shall define as the product of a relation (whose field consists of relations) a relation whose field is . Thus by the above proposition, whenever is a , the field of its product is the product (in the cardinal sense) of the fields of its field, just as the field of its sum is (by *162·22) the sum of the fields of its field.
*163·22.
Dem.
*163·3.
Dem.
*163·31.
*163·311.
Dem.
*163·32.
*163·33.
*163·331.
*163·4.
Dem.
*163·41.
Dem.
*163·42.
Dem.
The above proposition is used in *251·22.
*163·43.
Dem.
*163·431.
*163·44.
Dem.
*163·441.
The above proposition is used in *173·26.
*163·442.
Dem.
*163·45.
Dem.
*163·451.
The above proposition is used in *173·25.
*163·452.
*163·46.
*163·461.
*163·462.