Volume 1 / Part 5 — *201. TRANSITIVE RELATIONS.
Reading edition
*201. TRANSITIVE RELATIONS.
Summary of *201.
There are two main varieties of transitive relations, namely those that are symmetrical , and those that are asymmetrical . Transitive symmetrical relations have the formal properties of equality: examples of such relations have occurred above, e.g. identity, similarity, and likeness. The propositions of the present number, however, are rather such as will be useful in connection with transitive asymmetrical relations, since they are intended to be applied to series.
We denote the class of transitive relations by ""; thus Many propositions of this number are analogous to propositions whose numbers have the same decimal part in *200. Such are: If is transitive, so is its converse (*201·11), and so is any relation which is like P (*201·211); and are transitive (*201·3·31); if is transitive, so is (*201·33). The propositions *201·4—·42, which deal with the ideas of relation-arithmetic, are also analogous to *200·4—·42.
Most of the other propositions of this number, however, have no analogues in *200. Among the most important of these are the following :
*201·14.
*201·15.
*201·18.
This proposition is very important, since it effects an immense simplification in the use of all propositions involving or , when these propositions are to be applied to transitive relations. Owing to the above proposition, drops out where transitive relations are concerned. , on the other hand, remains useful: If , "" will mean " precedes or is ," which, if generates a series of which and are members, is equivalent to " does not follow ."
We have a series of propositions (*201·5—·56) on and . The chief of these are
*201·5.
*201·501.
These two propositions express the fact that a predecessor of a predecessor is a predecessor.
*201·52.
Thus if consists of together with the predecessors of its members.
*201·521.
*201·55.
We have next a set of important propositions on and . The chief are
*201·63.
*201·65.
On these two propositions, see the notes appended to them below.
*201·01.
*201·1.
*201·11.
Dem.
*201·12.
In virtue of this proposition, being contained in diversity is equivalent (where transitive relations are concerned) to asymmetry. This is not in general the case with relations which are not transitive; thus e.g. diversity itself is contained in diversity, but is symmetrical.
*201·13.
Dem.
*201·14.
Dem.
The following propositions (*201·15—·19) are concerned with and .
*201·15.
*201·16.
This proposition is important, since it often happens that a series is given as defined by a one-one relation , as in *122 for example, and in such cases is a serial relation in our present sense. By the above proposition, is always transitive; by *96·421, is connected when confined to the posterity of a given term, provided ; by *96·23, if and , is contained in diversity throughout the posterity of . Thus if is a one-one, confined to any family which has a beginning will be a serial relation.
*201·17.
Dem.
*201·18.
Dem.
This proposition is important, since it simplifies all propositions concerning and in case is transitive. The following proposition is an instance of this simplification.
*201·19.
The following propositions (*201·2—·22) are concerned in proving that transitiveness is unaffected by likeness-transformations, and therefore belongs to every member of a relation-number or to none.
*201·2.
Dem.
*201·201.
*201·21.
Dem.
*201·211.
This shows that transitiveness is a property which is unchanged by likeness-transformations. Hence
*201·212.
*201·22.
*201·3.
Dem.
*201·31.
Dem.
Unless , . A relation whose square is is transitive, because is contained in every relation.
*201·32.
Dem.
*201·33.
Dem.
The following propositions (*201·4—·42) are concerned with the ideas of relation-arithmetic.
*201·4.
Dem.
*201·401.
Dem.
*201·41.
Dem.
*201·411.
*201·42.
Dem.
The following propositions (*201·5—·56) are concerned with and , i.e. with the predecessors of some part of a class and the predecessors of the whole of a class.
*201·5.
*201·501.
*201·51.
Dem.
*201·52.
*201·521.
*201·53.
*201·54.
*201·55.
Dem.
The following proposition is a lemma which is used in *205·192 and *206·24.
*201·56.
Dem.
The following propositions, to the end of the number, are concerned with the relation defined in *121. We may regard as meaning "immediately precedes." *201·6·61·62 are lemmas for *201·63.
*201·6.
Dem.
*201·61.
Dem.
*201·26.
*201·63.
The above proposition is of fundamental importance. The relation (defined in *121) plays a great part in the theory of series. It is the relation "immediately preceding." Its domain consists of those terms which have immediate successors; its converse domain, of those that have immediate predecessors. In well-ordered series, , while consists of all terms (except the first) which do not belong to the first derivative (cf. *216). In any series, consists of all the terms which are limits of ascending series, and . consists of all the terms which are limits of descending series.
*201·64.
Dem.
*201·65.
When is a series, is the condition for its being a compact series, i.e. one in which there are terms between any two. In virtue of *201·65, this condition is equivalent to , which states that no term has an immediate predecessor.
The following proposition is first used in *253·521.
*201·66.
Dem.
*201·661.
Dem.
The above proposition is a lemma for the following.
*201·662.
This proposition is first used in *253·521.