Volume 1 / Part 5 — *205. MAXIMUM AND MINIMUM POINTS.
Reading edition
*205. MAXIMUM AND MINIMUM POINTS.
Summary of *205.
The minimum points of a class with respect to a relation are those members of which belong to the field of but to which no members of have the relation ; that is, they are those members of which belong to but have no predecessors in . Similarly the maximum points of are those members of which belong to but have no successors in . Both these notions have been already defined in *93, but they were there only used for the special purpose of studying generations. Their chief utility is in connection with series, and it is in this connection that we shall now consider them. Many of the properties of maxima and minima in series do not demand the whole hypothesis "," but only "." This is the case, in particular, with the fundamental property of maxima and minima in series, namely that each class has at most one maximum and one minimum. The minimum of a class, if it exists, is the first term of the class, and the maximum, if it exists, is the last term. The maxima with respect to are the minima with respect to ; hence properties of maxima result immediately from the corresponding properties of minima, and will be set down without proof in what follows.
It will be seen that the maxima and minima of depend only upon : the part of (if any) which is not contained in is irrelevant.
In accordance with the definitions of *93, the class of minima of is denoted by , where the definition being Thus is a relation contained in . When is connected, we have , i.e. (by *71·12) It follows that, if is a set of classes which all have minima, is a selective relation for , i.e. Owing to this fact, the existence of selections can sometimes be proved in dealing with series (especially with well-ordered series), in cases where such proof would be impossible if no serial arrangement were given.
The definition of is so chosen as to exclude from whatever part of is not contained in , and to make , i.e. , provided . For these two reasons we have to reject two simpler definitions which might otherwise be thought preferable. One of these would give which might be obtained by putting This agrees with our definition whenever , but not otherwise, since it includes in any part of not contained in . Hence it necessitates the hypothesis in many propositions which, with our definition, do not require this hypothesis, and in particular in the proposition so that instead of having (as with our definition) we should only have For these reasons, this definition is less convenient than the one we have adopted.
The other definition which suggests itself is one which will give If this definition were adopted, we might dispense with a special notation altogether, using , in place of , . This definition, however, has the drawback that, if and , so that we have This necessitates the addition of the hypothesis (as in *204·45 above, for example) in cases where, with our definition, no such hypothesis is required. If we take , instead of , as the class of minimum points, we secure when and , but not when . Thus we still have exceptions to provide against which do not arise with the definition we have adopted.
The first few propositions of this number have already been proved in *93, but are repeated here for convenience of reference.
The propositions of this number are numerous and much used. Among the elementary properties of and with which the number begins, the following should be noted:
*205·12.
*205·123.
*205·14.
*205·15.
*205·16.
*205·18.
*205·19.
*205·194.
Owing to this proposition, we can sometimes dispense with the hypothesis in propositions about minima which would otherwise require this hypothesis.
*205·197.
Our next set of propositions (*205·2—·27) introduces the hypothesis that is connected, or transitive and connected. The chief of them are
*205·21.
I.e. if the minimum of exists, it precedes every other member of .
*205·22.
I.e. the terms which come after some part of are those that come after its minimum (when the minimum exists).
*205·25.
We have next the fundamental proposition:
*205·3.
whence
*205·31.
which leads to
*205·33.
This proposition is useful in the theory of well-ordered series. Observe that "" means that consists of classes which have minima.
We have next a set of propositions (*205·4—·44) dealing with the relations of to and ; next we have propositions on the relations of the minima of two different classes, of which the most useful is
*205·55.
We have next various propositions on , of which the chief is
*205·65.
I.e. the predecessors of the whole of a class contained in are the predecessors of its minimum (if it has one).
A useful proposition is
*205·68.
I.e. if is a hereditary class, its minima with respect to are the same as its minima with respect to .
We prove next that if has a maximum, so has (*205·7), and that if , only a unit class can have its maximum identical with its minimum (*205·73).
*205·8—·85 are concerned with relation-arithmetic. The chief proposition here is
*205·8.
I.e. in any correlation, the minima of the correlates of a class are the correlates of the minima.
We end with two propositions on relations with limited fields. The more useful of these is
*205·9.
*205·1.
*205·101.
*205·102.
*205·11.
*205·111.
*205·12.
*205·121.
*205·122.
*205·123.
*205·13.
*205·131.
*205·14.
*205·141.
*205·15.
*205·151.
*205·16.
*205·161.
*205·17.
Dem.
*205·18.
Dem.
*205·181.
Dem.
*205·182.
Dem.
*205·183.
Dem.
*205·19.
Dem.
*205·191.
*205·192.
Dem.
*205·193.
*205·194.
Dem.
*205·195.
*205·196.
Dem.
*205·197.
*205·2.
Dem.
In the remainder of the present number, when a proposition has been proved for , we shall not state the corresponding proposition for , unless it is specially important. When propositions concerning are required for reference in the sequel, we shall refer to the corresponding propositions for , in case no reference exists for .
*205·*21.
*205·211.
Dem.
*205·22.
*205·23.
Dem.
*205·24.
*205·241.
*205·25.
Dem.
The following proposition is used in the theory of well-ordered series (*250·2).
*205·251.
*205·252.
*205·253.
*205·254.
*205·255.
Dem.
*205·256.
*205·26.
Dem.
*205·261.
Dem.
*205·262.
Dem.
*205·27.
Dem.
The above proposition is used in *250·7.
*205·3.
Dem.
The above proposition is of great importance in the theory of maxima and minima.
*205·31.
*205·32.
*205·33.
Dem.
*205·34.
The following proposition is used in *260·17.
*205·35.
Dem.
*205·36.
Dem.
The above proposition is used in *230·53.
*205·37.
The following proposition is used in *257·21.
*205·38.
Dem.
*205·381.
Dem.
The three following propositions lead up to *205·42, which is used in *261·26.
*205·4.
Dem.
*205·401.
Dem.
The following proposition, besides being required for *205·41, is used in *250·151.
*205·41.
Dem.
*205·42.
Dem.
The following proposition leads up to *205·44.
*205·43.
Dem.
*205·44.
The following propositions deal with the circumstances under which the minimum of one class is identical with, or earlier than, that of another.
*205·5.
Dem.
*205·501.
Dem.
*205·51.
Dem.
*205·52.
Dem.
*205·53.
Dem.
*205·54.
*205·55.
Dem.
*205·56.
Dem.
*205·561.
*205·6.
*205·601.
*205·61.
*205·62.
*205·63.
*205·64.
Dem.
*205·65.
Dem.
*205·66.
*205·67.
Dem.
*205·68.
Dem.
*205·681.
*205·7.
Dem.
*205·71.
Dem.
*205·72.
*205·73.
Dem.
*205·731.
*205·732.
Dem.
The following propositions lead up to *205·75, which shows that the minimum of a class belongs to unless the part of the class contained in is .
*205·74.
Dem.
*205·741.
Dem.
*205·742.
Dem.
*205·75.
Observe that is not in general equivalent to , since the latter implies , while the former does not.
The following proposition is important.
*205·8.
Dem.
*205·81.
Dem.
*205·82.
The two following propositions are used in *251·13.
*205·83.
Dem.
*205·831.
Dem.
The two following propositions are used in *251·14.
*205·832.
Dem.
*205·833.
Dem.
The following proposition is used in *251·25.
*205·84.
Dem.
*205·841.
Dem.
The following proposition is used in *251·2.
*205·85.
Dem.
*205·9.
*205·91.
Dem.