Volume 1 / Part 1 — *34. THE RELATIVE PRODUCT OF TWO RELATIONS.
Reading edition
*34. THE RELATIVE PRODUCT OF TWO RELATIONS.
Summary of *34.
The relative product of two relations and is the relation which holds between and when there is an intermediate term such that has the relation to and has the relation to . Thus e.g. the relative product of brother and father is paternal uncle; the relative product of father and father is paternal grandfather; and so on. The relative product of and is denoted by ""; the definition is:
*34·01.
This definition is only significant when and belong to the same type.
The relative product of and is called the square of ; we put
*34·02.
*34·03.
The most useful propositions in the present number are the following:
*34·2.
I.e. the converse of a relative product is obtained by turning each factor into its converse and reversing the order of the factors.
*34·21.
I.e. the relative product obeys the associative law.
*34·25.
*34·26.
I.e. the relative product obeys the distributive law with respect to the logical addition of relations. (For logical multiplication instead of logical addition, we only get inclusion instead of identity; cf. *34·23·24.)
*34·34.
*34·36.
*34·41.
*34·01.
*34·02.
*34·03.
*34·1.
*34·11.
Dem.
*34·12.
*34·2.
Dem.
*34·202.
Dem.
*34·203.
*34·21.
Dem.
*34·22.
This definition serves merely for the avoidance of brackets.
*34·23.
Dem.
The converse of the above is not true.
*34·24.
*34·25.
Dem.
*34·26.
The above two forms of the distributive law, and the associative law (*34·21), are the only ones of the usual formal laws that hold for the relative product. The commutative law, in particular, does not hold in general.
*34·27.
Dem.
*34·28.
*34·29.
Dem.
In proving the equality of two relations, say and , we usually establish first an asserted proposition of the form
We then proceed by *11·11 (together with *11·3 in the second case) to whence the result follows by *21·43. We shall in future omit these steps, and write "" after we have established A similar ellipsis will be made in proving the equality of classes.
*34·3.
Dem.
*34·301.
*34·302.
Dem.
*34·31.
Dem.
*34·32.
*34·33.
Dem.
*34·34.
Dem.
*34·35.
Dem.
*34·351.
*34·36.
Dem.
The following proposition is a lemma for *95·31.
*34·361.
Dem.
*34·37.
*34·38.
*34·4.
Dem.
*34·41.
Dem.
The above proposition is no longer true if we change the hypothesis into , since ( may exist when does not. Suppose, e.g., that is the relation of child to father, and the relation of daughter to father. Then (= the granddaughter of , but = the daughter of the child of . The first exists whenever has only one granddaughter, while the second requires further that should have only one child.
For the same reason we do not have This will hold if , are one-many relations (cf. *71), but not in general otherwise.
*34·42.
Dem.
*34·5.
*34·51.
Dem.
*34·52.
*34·53.
*34·531.
*34·54.
Dem.
*34·55.
*34·56.
*34·6.
Dem.
*34·62.
Dem.
The above proposition is a lemma for *160·51, as is also *34·73, which employs the above proposition.
*34·63.
Dem.
*34·7.
Dem.
Thus is always a symmetrical relation, i.e. one which is equal to its converse.
*34·701.
*34·702.
Dem.
*34·703.
*34·73.
Dem.
*34·8.
Dem.
The hypothesis of the above proposition is the hypothesis that is symmetrical () and transitive (. These are the formal properties of those relations which can suitably be regarded as expressing equality in some respect.
*34·81.
The following propositions are lemmas for *34·85, which is used in *72·64:
*34·82.
Dem.
*34·83.
Dem.
*34·84.
Dem.
*34·841.
Dem.
*34·85.