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

574

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

Then our series consists of (PQ)'-R. The sum of all relations of this class
is considered in this number.
The principal propositions proved in *94 and *95 are two which have the
same hypothesis as the Schroder-Bernstein theorem, namely
R, S e 1 -1. I'S C D'R. a'R C D'S.
These two propositions state that, with the above hypothesis,
s'gen'(R I S) sm s'gen'(S I R)
and p'(I"Pot'(R I S) sm p'(I"(S I R).
The two combined reconstitute the Schr6der-Bernstein theorem, since
s'gen'(R S) up'CI"Pot'(R S) = D'R
and s'gen'(S R) v p'("Pot'(S R) = DS.
Thus they present, so to speak, an itemized account of the equality proved
by the Schroder-Bernstein theorem.
4 -*96, on the posterity of a term, is concerned with the properties of R*'x,
4 -chiefly when R e Cls -- 1. In this case, in general, R*'x consists of two parts,
first an open series and then a cyclic series. Either of these may vanish, or
may reduce to a single term. If we call the two parts 3 and y, the whole of
i3 precedes the whole of y, and 31 R, 1 R 1R- 1. Thus if either i3 or 7
4 -vanishes, Rm'x 1 R e 1. If ry vanishes, the series never returns into
itself, that is, R*'x 1 Ro C J. If y exists, there is a definite power of R, say
T, such that y e y. y. yTy. If 3 and y both exist, there is one term, namely
the successor of the last term of /, which has just two immediate predecessors,
4 —
one in /3 and one in y; every other term of Ro'x has only one immediate
4- 4 -predecessor in R*'x. Thus R*'x is shaped like a Q, with x at the tip of the
tail.
*97 deals with the analysis of the field of a relation into families. Taking
- 4 -any member x of C'R, the family of x with respect to R is R*'x u R'Cx, which
we write R*'x. Thus the class of families is R*"C'R. Those families which
4- v -
contain a member of B'R are R*"B'R. If we regard RB"B'R as arranged
in a rectangle, in which the generations are the successive rows, then
R*"B'R will be the columns. Thus the relation of geu'R to R*"B'R may
be regarded as a generalized form of the relation of rows and columns.
Under a suitable hypothesis, each row is a selection from the columns, and
each column a selection from the rows. This is expressed in the following
proposition: