Volume 1 / Part 3 — *114. THE ARITHMETICAL PRODUCT OF A CLASS OF CLASSES.
Reading edition
*114. THE ARITHMETICAL PRODUCT OF A CLASS OF CLASSES.
Summary of *114.
The kind of multiplication defined in *113 cannot be extended beyond a finite number of factors. We therefore, as in the case of addition, introduce another definition, defining the product of the numbers of a class of classes, and capable of being applied to an infinite number of factors. We define the product of the numbers of members of as ; thus we put
It is to be observed that is not a function of , because, if two members of have the same number, this will count only once in , but will count twice in .
It is very easy to see that, in case is finite, will be what we should ordinarily regard as the product of the numbers of members of . For suppose (e.g.) where . Then
Thus if is a member of , is determinate when , , are given, , , being the referents to , , . Whether , , overlap or not, the choice of any one of , , is entirely independent of the choice of the other two, and therefore the total number of choices possible is obviously the product of the numbers of , , . Thus our definition will not conflict with what is commonly understood by a product.
The propositions of this number are less numerous and less important than those of *113. We shall deal first with products of a single factor, and products in which one factor is null (*114·2—·27). We shall then deal (*114·3—·36) with the relations between the sort of multiplication here defined and the sort defined in *113. Then we have a few propositions (*114·4—*114·43) showing that unit factors make no difference to the value of a product. Then we prove (*114·5—·52) that the value of the product is the same for two classes having double similarity, and then (*114·53—·571) we give extensions of this result which depend upon the multiplicative axiom. Finally, we give some new forms of the associative law of multiplication.
Among the more important propositions in this number are the following:
*114·21.
I.e. a product of one factor is equal to that factor.
*114·23.
I.e. a product vanishes if one of its factors is zero. The converse requires the multiplicative axiom, as appears from the proposition
*114·26.
I.e. the multiplicative axiom is equivalent to the assumption that a product vanishes when, and only when, one of its factors is zero.
*114·301.
whence
*114·31.
which is a form of the associative law, and
*114·35.
which connects the two sorts of multiplication.
*114·41.
I.e. unit factors make no difference to the value of a product.
*114·51.
This proposition gives a correlator of and as a function of a double correlator of and , and thus leads to
*114·52.
Hence, by the propositions of *111, we infer
*114·571.
I.e. assuming the multiplicative axiom, if and each consist of classes of terms each, their products are equal.
We have next various forms of the associative law, beginning with
*114·6.
which is an immediate consequence of *85·44. The other form is
*114·632.
As to the sense in which this is a form of the associative law, see the observations following *114·6.
*114·01.
*114·1.
*114·11.
*114·12.
*114·2.
Thus a product of no factors is 1. This is the source of , as we shall see later.
*114·21.
*114·22.
*114·23.
Thus an arithmetical product is zero if any of its factors is zero. To prove the converse, we have to assume the multiplicative axiom, which, in fact, is equivalent to the proposition that an arithmetical product is only zero when at least one of its factors is zero.
*114·24.
Dem.
*114·25.
Dem.
Note that .
*114·26.
*114·261.
*114·27.
*114·3.
Dem.
*114·301.
Dem.
*114·31.
The above is one form of the associative law of multiplication.
*114·311.
*114·32.
Dem.
*114·33.
*114·34.
*114·35.
*114·36.
*114·4.
*114·41.
*114·42.
Dem.
*114·43.
*114·5.
Dem.
*114·501.
Dem.
*114·51.
*114·52.
*114·53.
*114·54.
The condition , , which is involved in the hypothesis of *114·54 (through , ), is not necessary. The following propositions enable us to remove it. We first prove and then we use *114·54 to take us from to . Thence we arrive at .
*114·56.
*114·561.
*114·562.
Dem.
*114·57.
Dem.
*114·571.
*114·6.
This is the most general form of the associative law for arithmetical multiplication.
Owing to the fact that we have two kinds of multiplication, namely and , we have four forms of the associative law of multiplication, namely:
(1) *114·6, above,
(2) *113·54, i.e. ,
(3) *114·31, i.e. ,
(4) a form of the associative law which has not yet been proved, which may be explained as follows.
Suppose we have a number of pairs of classes, e.g. , , ,.... Suppose we form the products , , , ... and multiply all these products together. We wish to prove that (with a suitable hypothesis) the result is similar to the product of all the 's and all the 's taken together as one class; i.e. if we call the class of products , , , ..., and the class whose members are , , , ..., , , , ..., we wish to prove In order to express this proposition in symbols, let be the correlator of the 's and 's, so that . (The suffix will not be used further, since it implies that the number of 's and of 's is finite or denumerable.) Then our class of products of the form is where is the class of all the 's ; and the product of this class of products is On the other hand, the class of all the 's and 's is , and the product of this class is Thus what we have to prove (with a suitable hypothesis) is The hypothesis required is
A smaller hypothesis suffices, however, for a proposition which, in virtue of *114·301, is closely allied to the above, namely For this, a sufficient hypothesis is Thus e.g. we may write for , and we find
We shall now prove the above propositions. What follows, down to *114·621, consists of lemmas.
For convenience, we write for in the course of these lemmas; this notation is introduced in the hypotheses of the lemmas.
*114·601.
Dem.
*114·602.
Dem.
As in *114·601, we prove
*114·603.
Dem.
*114·604.
The relation here defined is the correlator required for proving
Besides what is proved in the present proposition, we shall have to prove
The proof of the present proposition is as follows.
Dem.
*114·605.
Dem.
The following propositions are required for proving that, with the same hypothesis, .
*114·61.
Dem.
*114·611.
Dem.
*114·612.
Dem.
*114·613.
Dem.
*114·614.
Dem.
*114·62.
*114·621.
The hypothesis is not necessary, since, when ,
are both . This is proved in *114·63.
*114·63.
Dem.
The above is one of the two variants of the associative law for and .
*114·631.
*114·632.
This is the second variant of the associative law for and .
*114·64.
Dem.
In the above proposition, the hypothesis has to be such as to yield . Various other forms of hypothesis will secure this result, and will give other forms of the above proposition. This subject is treated in *74, above.
*114·65.