Volume 1 / Part 1 — SECTION E. PRODUCTS AND SUMS OF CLASSES.

Summary of Section E.

In the present section, we make an extension of , , , . Given a class of classes, say , the product of (which is denoted by is the common part of all the members of , i.e. the class consisting of those terms which belong to every member of . The definition is If has only two members, and say, . If has three members, , , , then ; and so on. But this process can only be continued to a finite number of terms, whereas the definition of does not require that should be finite. This notion is chiefly important in connection with the lower limits of series. For example, let be the class of rational numbers whose square is greater than 2, and let "" mean ", where and are rationals." Then if , will be the class of rationals less than x. Thus will be the class of such classes as , where . Thus the product of , which we call , will be the class of rationals which are less than every member of , i.e. the class of rationals whose squares are less than 2. Each member of is a segment of the series of rationals, and is the lower limit of these segments. It is thus that we prove the existence of lower limits of series of segments.

Similarly the sum of a class of classes is defined as the class consisting of all terms belonging to some member of ; i.e. i.e. belongs to the sum of if belongs to some . This notion plays the same part for upper limits of series of segments as plays for lower limits. It has, however, many more other uses than , and is altogether a more important conception. Thus in cardinal arithmetic, if no two members of have any term in common, the arithmetical sum of the numbers of members possessed by the various members of is the number of members possessed by .

The product of a class of relations ( say) is the relation which holds between and when and have every relation of the class . The definition is The properties of are analogous to those of , but its uses are fewer.

The sum of a class of relations ( say) is the relation which holds between and whenever there is a relation of the class which holds between and . The definition is This conception, though less important than , is more important than . The summation of series and ordinal numbers depends upon it, though the connection is less immediate than that of the summation of cardinal numbers with .

Instead of defining , , , , it would be formally more correct to define , , and , which are the relations giving rise to the above descriptive functions. Thus we should have whence we should proceed to

But in cases where the relation, as opposed to the descriptive function, is very seldom required, it is simpler and easier to give the definition of the descriptive function in the first instance. In such cases, the relation is always tacitly assumed to be also defined; i.e. when we give a definition of the form where is some previously defined relation, we always assume that this definition is to be regarded as derived from

In addition to products and sums, we deal, in the present section, with certain properties of the relations and , the meanings of which result from the notation introduced in *38. Such relations are very useful in arithmetic. The reason for dealing with them in the present section is that a large proportion of the propositions to be proved involve sums of classes of classes or relations.