Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*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 x, y e C'P. x y,
we have xPy. v. yPx. 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 X is a class of transitive or
asymmetrical relations, s'X is transitive or asymmetrical; but if X is a class
of connected relations, s'X is not in general connected. Hence if X is a class
of series, s'X 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 S'C'P,
but as s'C'P v F;P (cf. *162), because the latter, but not in general the former,
is connected when P and all the members of CYP are connected (*202'42).
When P is connected, if a is any class contained in C'P, we have
C'P = P"a v av (C'P npP a),
and there is at most one member of a belonging neither to P"a nor to
4 -C'Pnp'P"a. This member of a, if it exists, is the maximum of a. If,
4 -further, P2 C J (i.e. if P is asymmetrical), (P"a v a) n (C'P nrp'P"a)= A.
Thus when P is both connected and asymmetrical, P"a v a and C'P nr p'P"a
are each other's complements, and the two together constitute the Dedekind
cut defined by a, P"a v a being all the terms that do not follow the whole of
a, and C'P n p'P"a being all the terms that do follow the whole of a.
More generally, if a is any class, not necessarily contained in C'P, then
when P is connected, we have
C'P -p'P"(a n C'P) C P"a v (a n C'P),
and when P is asymmetrical, we have
4 -P"a v (a nr CP) C C'P - p'P"a.
Thus when both conditions are fulfilled, we have (*202'503)
C'P - p'P"(a n C'P) = P"a v (a n C'P).