Volume 1 / Part 4 — *174. THE ASSOCIATIVE LAW OF RELATIONAL MULTIPLICATION.
Reading edition
*174. THE ASSOCIATIVE LAW OF RELATIONAL MULTIPLICATION.
Summary of *174.
In the present number, we have to prove the associative law for and for , i.e. we have to prove (with a suitable hypothesis) and .
The first of these requires and either or the second requires not only this, but also . When both and are relations of mutually exclusive relations, we call an arithmetical relation, which we denote by "." Arithmetical relations serve exactly analogous purposes to those served by arithmetical classes in cardinal arithmetic.
The proof of the associative law for consists in showing that, under a suitable hypothesis, (with its converse domain limited) is a correlator of and (*174·221·23). To prove this, we first prove
*174·17.
and
*174·19.
This gives what we may call the cardinal part of the proof, i.e. it shows that is a cardinal correlator of the fields of and . We then prove that if and belong to the field of , they have the relation when the relational sums of their domains have the relation . Here, in addition to the hypothesis , we require that if any relation has the relation to itself, then is not to have more than one term. Thus we have
*174·215.
The hypothesis is verified if (*174·216); thus for most purposes it is more convenient to substitute the simpler hypothesis for . We shall, however, have occasion to use the hypothesis in *182·42·43·431, where our is a relation whose field consists entirely of relations of the form , whose fields are always unit classes, so that our satisfies the above hypothesis even if is not contained in .
The proof of *174·215 (above) is effected by first proving
*174·2.
From *174·17·19·215 we deduce
*174·221.
whence we obtain the more convenient proposition
*174·23.
Thus if the hypothesis of *174·221 or of *174·23 holds, the associative law holds for (*174·241·25).
To prove the associative law for , i.e. we observe that, since (*174·23) we have (*174·41) Also , by *115·46. Hence the associative law follows (*174·43). It will be observed that in this case the correlator is simply s with its converse domain limited (*174·42).
As in the case of , "" is a stronger hypothesis than we really need: what we need is .
*174·01.
*174·12.
Dem.
*174·13.
*174·16.
Dem.
*174·161.
Dem.
*174·162.
Dem.
*174·17.
Dem.
*174·18.
Dem.
*174·19.
Dem.
*174·191.
Dem.
*174·2.
Dem.
*174·21.
Dem.
*174·211.
Dem.
*174·212.
*174·213.
Dem.
*174·214.
Dem.
*174·215.
Dem.
*174·216.
Dem.
*174·22.
*174·221.
Dem.
*174·23.
*174·231.
Dem.
*174·24.
*174·241.
*174·25.
This proposition gives the associative law for . It remains to prove the associative law for .
The following propositions are concerned with various properties of "arithmetical" relations, down to *174·4, where the proof of the associative law for begins.
*174·3.
*174·31.
*174·311.
*174·32.
*174·321.
*174·322.
Dem.
*174·33.
Dem.
*174·34.
Dem.
*174·35.
Dem.
*174·36.
Dem.
*174·361.
Dem.
*174·362.
Dem.
*174·363.
Dem.
*174·4.
Dem.
*174·401.
Dem.
*174·41.
*174·42.
Dem.
*174·43.
This is the associative law for .
*174·44.
*174·45.
Dem.
*174·46.
*174·461.
*174·462.
The two following propositions merely sum up previous results.
*174·47.
*174·48.