Volume 1 / Part 5 — *210. ON SERIES OF CLASSES GENERATED BY THE RELATION OF INCLUSION.
Reading edition
*210. ON SERIES OF CLASSES GENERATED BY THE RELATION OF INCLUSION.
Summary of *210.
In the theory of series it frequently happens that we have to deal with a class of classes such that, of any two, one is contained in the other. I.e. if is the class of classes, we have
Instances of this are afforded by the various classes of sections, to be considered in *211. When fulfils the above condition, the classes composing can be arranged in a series by the relation of inclusion (combined with inequality), i.e. by the relation or, what comes to the same, If is any relation such that , the above relation of inclusion is equal to (For the definition of , see *170.) Thus under the above circumstances, is a series, whatever may be.
The importance of such relations of inclusion, as generators of series, is in connection with the existence of maxima and minima or limits. If we put where satisfies the above condition, then if and if , is the maximum or the upper limit of with respect to , according as is a member of or not. Similarly if , p is the minimum or lower limit of , according as is a member of or not. Hence if is such that the sum of any sub-class of is a member of , every sub-class of has either a maximum or an upper limit; and if the product of every sub-class of is a member of , every sub-class of has either a minimum or a lower limit.
In order that every sub-class of should have a minimum or a lower limit, it is sufficient that the sum of every sub-class of should be a member of . For, if is any sub-class of , consider those members of which are contained in , i.e. If , the sum of these classes = , and is the lower limit or minimum of . But if , then every member of which is not contained in is also not contained in , and is therefore not contained in some member of . Hence is the lower limit of .
It is owing to these propositions that segments of series are of such great importance in connection with limits.
The hypothesis that if , is a member of , will usually fail to be verified in the case when , since in this case . But all the results desired can be obtained from the hypothesis that, if , . This hypothesis is equivalent to the other except in the case of , in which case it requires , which is much more often verified than , which was required by the other hypothesis.
The principal propositions of this number are the following:
*210·1.
*210·11.
*210·12.
*210·13.
*210·2.
*210·21.
*210·211 gives an analogous proposition for and . We shall not here mention such analogues, unless for some special reason.
*210·23.
*210·232.
*210·251.
*210·252.
*210·254.
*210·26.
*210·28.
Thus if is a class of not less than two classes such that, of any two of its members, one must be contained in the other, and if is the relation confined to members of , then is a series (*210·12) in which, provided the sums of sub-classes of are always members of , every class has either a maximum or an upper limit, and every class has either a minimum or a lower limit (*210·28).
The reader will observe that, if , , any finite sub-class of must contain its own sum and product as members. For example, if we have two classes and , if , then and ; if we have three classes , , , and , then and ; and so on. Thus the hypothesis is only required in order to enable us to deal with infinite sub-classes of .
*210·1.
Dem.
*210·11.
Dem.
*210·12.
Dem.
*210·121.
Dem.
*210·122.
Dem.
*210·123.
Dem.
*210·124.
*210·13.
Dem.
Thus under the hypothesis of *210·1, does not depend upon , so long as . Also we have
*210·14.
*210·15.
*210·16.
Dem.
*210·17.
Dem.
*210·2.
Dem.
Observe that is either or , according as is or is not a member of .
*210·201.
*210·202.
*210·203.
*210·21.
*210·211.
*210·22.
*210·221.
*210·222.
*210·223.
*210·23.
Dem.
*210·231.
In virtue of *210·21·23, every class which is contained in , and whose product is a member of , has either a minimum or a lower limit; and in virtue of *210·211·231, every class which is contained in , and whose sum is a member of , has either a maximum or an upper limit.
*210·232.
*210·233.
*210·24.
*210·241.
*210·242.
*210·25.
Dem.
*210·251.
*210·252.
Dem.
This proposition is more useful than *210·25, because its hypothesis is much oftener verified. In order that the hypothesis of *210·25 may be verified, we must have , since ; hence we must also have . But the hypothesis of *210·252 only requires, as far as is concerned, that we should have .
*210·253.
*210·254.
*210·26.
Dem.
*210·261.
*210·262.
Dem.
The same remark applies to this proposition as to *210·252.
*210·27.
Dem.
*210·271.
*210·272.
*210·28.
Dem.
*210·281.
*210·282.
Thus when either of the hypotheses of *210·281·282 is fulfilled, the series is Dedekindian both upwards and downwards.
*210·29.
*210·291.