Volume 1 / Part 5 — *202. CONNECTED RELATIONS.
Reading edition
*202. CONNECTED RELATIONS.
Summary of *202.
A relation is said to be connected when either it or its converse holds between any two different members of its field, i.e. when, if , , we have . Thus the field of a connected relation consists of a single family, unless the relation is null, in which case it has no families. Conversely, a relation which has one family or none is connected. Connection is necessary, in addition to transitiveness and asymmetry, in order that a relation may generate a single series. If is a class of transitive or asymmetrical relations, is transitive or asymmetrical; but if is a class of connected relations, is not in general connected. Hence if is a class of series, is not one series, but many detached series. This is one reason why the arithmetical sum of a relation of relations is not defined as , but as (cf. *162), because the latter, but not in general the former, is connected when and all the members of are connected (*202·42).
When is connected, if is any class contained in , we have and there is at most one member of belonging neither to nor to . This member of , if it exists, is the maximum of . If, further, (i.e. if is asymmetrical), . Thus when is both connected and asymmetrical, and are each other's complements, and the two together constitute the Dedekind cut defined by being all the terms that do not follow the whole of , and being all the terms that do follow the whole of .
More generally, if is any class, not necessarily contained in , then when is connected, we have and when is asymmetrical, we have Thus when both conditions are fulfilled, we have (*202·503)
The above inclusions and the consequent equality will be constantly required throughout what follows. The division of into the two mutually exclusive parts is the Dedekind "cut" defined by the class . If , the two parts become, as above mentioned, If, further, is not null, they become If is contained in and contains all its own predecessors, they become In this simplified form, Dedekind "cuts" will be considered later (*211).
We take as our definition
Some of the propositions of the present number are analogues of propositions in *200 and *201. Such are: If is connected, so is (*202·11); if is connected, so is any similar relation (*202·211); and are connected (*202·3·31); if is connected, so is (*202·33); and various propositions connected with relation-arithmetic (*202·4—·42). The majority of the propositions of this number, however, deal with properties peculiar to connexity. Among the most important of these are:
*202·101.
*202·103.
These are merely alternative forms of the definition.
*202·13.
*202·5.
*202·501.
*202·503.
*202·505.
*202·52.
*202·524.
*202·55.
In virtue of this proposition (and others) if is a series and is a class (not a unit class) contained in , is the generating relation of the series consisting of the class in the order which it has in the series .
*202·7.
This proposition is to be taken in connection with *201·63. The two together show that when is a series, is one-one.
*202·01.
For the definition of , see *97·01.
*202·1.
*202·101.
*202·102.
*202·103.
*202·104.
*202·11.
*202·12.
Dem.
The following propositions, down to *202·181 inclusive (excepting *202·16·161) are concerned with and . It often happens that these are connected when is not so, e.g. if is the relation among inductive cardinals.
*202·13.
Dem.
*202·131.
*202·132.
*202·133.
Dem.
*202·134.
*202·135.
Dem.
*202·136.
*202·137.
*202·138.
*202·14.
*202·141.
*202·15.
Dem.
The above proposition is used in the ordinal theory of finite and infinite (*260·4).
*202·16.
Dem.
*202·161.
Dem.
*202·162.
Dem.
*202·17.
Dem.
*202·171.
*202·172.
*202·18.
Dem.
*202·181.
Dem.
The above proposition is used in the ordinal theory of finite and infinite (*261·2).
The following proposition is a lemma for *202·211, which shows that if a relation is connected, so are all similar relations.
*202·21.
Dem.
The proofs of the three following propositions proceed like the proofs of the analogous propositions in *200 and *201.
*202·211.
*202·212.
*202·22.
*202·3.
Dem.
*202·31.
Dem.
*202·33.
Dem.
The following propositions (*202·4—·42) are concerned with applications of relation-arithmetic.
*202·4.
Dem.
The above proposition illustrates the reasons for defining as was done in *160. When and are connected, is in general not connected: it is the additional term which insures connection.
*202·401.
Dem.
*202·41.
Dem.
*202·411.
*202·412.
Dem.
*202·42.
Dem.
*202·5.
Dem.
The following propositions (*202·501-·51) are concerned with the relations of and . They are important, and *202·501·503·505 will be often used.
*202·501.
Dem.
*202·502.
Dem.
*202·503.
Dem.
*202·504
Dem.
*202·505.
Dem.
*202·51.
Dem.
The following propositions (*202·511—·524) are concerned with . *202·52 shows that if , cannot have more than one first term or more than one last term, and *202·523 shows that this still holds if only is connected. *202·511 shows that if is a connected relation which has a first term, then if is any class, there are predecessors of the whole of when and only when is such a predecessor, and when and only when . *202·524 shows that if is connected and has a first term, consists of the successors of the first term. These propositions are much used.
*202·511.
Dem.
*202·52.
Dem.
*202·521.
Dem.
*202·522.
*202·523.
*202·524.
Dem.
The following propositions (*202·53—·55) are concerned with relations with limited fields. Such relations are constantly used in the theory of series.
*202·53.
Dem.
This proposition is important in series. If and are serial relations, and , they verify the above hypothesis; hence if is a series contained in a given series , is simply with its field limited. Thus series contained in a given series are completely determined by their fields.
*202·54.
Dem.
The above proposition is frequently used. *202·55, which is an immediate consequence of *202·54, is used incessantly.
The following proposition is used in *232·14.
*202·541.
Dem.
*202·55.
*202·56.
Dem.
The above proposition is used in *212·652.
*202·6.
Dem.
The following proposition is a lemma for *202·62, which is itself a lemma for *204·52.
*202·61.
Dem.
*202·611.
*202·62.
Dem.
The three following propositions (*202·7—·72) are concerned with . Of these, *202·7 is important: it shows that if is connected, no term can have more than one immediate predecessor or successor. *202·72 is used in *204·71, which is an important proposition.
*202·7.
Dem.
*202·71.
Dem.
*202·72.
*202·8.
Dem.
*202·81.
Dem.
The above proposition shows that if is connected, and any class is picked out of , then arranges in an order which is similar to that in which arranges the correlates of .