Volume 1 / Part 4 — *162. THE SUM OF THE RELATIONS OF A FIELD.
Reading edition
*162. THE SUM OF THE RELATIONS OF A FIELD.
Summary of *162.
The form of summation defined in *160 cannot be extended beyond a finite number of summands, since it involves explicit mention of all the summands. In the present number, we shall be concerned with a form of summation which is not subject to this restriction. It will be observed that, since relational summation is not permutative, we cannot define the sum of a class of relations, for this would not determine the order in which the summation is to be effected. Our relations must be given as the field of some relation which orders them; thus the sum appears not as the sum of a class, but as the sum of a relation, namely of a relation whose field is the relations to be summed. In the case of two relations and , the sum of , as defined in the present number, will be equal to ; similarly for three, the sum of will be equal to , and so on for any finite number of summands.
As explained in the introduction to this Section, if P is a relation between relations, we put
It is convenient to suppose that is serial, and that every member of is also serial. Then holds between and if either (1) there is a series, in the field of , in which precedes , or (2) belongs to a series which is earlier, in the -series, than the series to which y belongs. The following are the chief propositions of this number:
*162·22·23.
*162·26.
*162·3.
*162·31.
*162·34.
*162·35.
This is the analogue of *40·38. (Cf. note to *162·35, below.)
*162·4.
*162·42.
*162·43.
It should be observed that the ordinal analogues of propositions about classes of classes often involve the substitution of (not ) for . Examples are afforded by *162·34·35, quoted above.
*162·01.
*162·1.
*162·11.
*162·12.
*162·13.
*162·14.
*162·2.
Dem.
*162·21.
Dem.
*162·211.
*162·212.
Dem.
*162·213.
The above proposition is used in *163·22.
The two following propositions are used very often.
*162·22.
Dem.
*162·23.
*162·26.
Dem.
*162·27.
*162·3.
Dem.
This proposition establishes the connection between the two kinds of arithmetical addition of relations.
*162·31.
Dem.
The following propositions lead up to *162·34.
*162·32.
Dem.
*162·33.
Dem.
*162·331.
Dem.
*162·332.
Dem.
*162·34.
This is the associative law for arithmetical sums of relations.
The following propositions lead up to *162·35.
*162·341.
Dem.
*162·342.
Dem.
*162·343.
Dem.
*162·35.
Dem.
This proposition is important, since it enables us to infer (with a suitable hypothesis) that if is always like when , then the arithmetical sum of all such relations as is like , being in fact . In other words, if, whenever , is a correlator of and , then is a correlator if and . This proposition is analogous in its uses to the proposition which is *40·38. In general, in obtaining relational analogues of cardinal propositions, is to be replaced by , by , and by . When these substitutions are made in , *162·35 results, except for its hypothesis.
If we regard as a kind of product of and , *162·35 becomes a distributive law. For it asserts that if we multiply each member of by , and then sum the resulting products, we get the same relation as if we first sum , and then multiply by . The following application of *162·35 to the sum of two relations makes its distributive character more evident.
*162·36.
Dem.
This proposition can be extended to any finite number of summands.
*162·37.
Dem.
*162·371.
*162·372.
*162·4.
Dem.
*162·41.
Dem.
*162·42.
Dem.
*162·43.
Dem.
*162·431.
Observe that in *162·43·431, and must be of different types, in fact must be of the type to which members of belong. *162·43·431 are often useful.
*162.44.
Dem.
*162·45.
Dem.
The above proposition is used in *174·162.