*34. THE RELATIVE PRODUCT OF TWO RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
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.