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

SECTION D]

NOTE TO SECTION D

315

of a relation which gives rise to a descriptive function without limiting its
domain or converse domain consists of all possible values of the function;
the converse domain consists of all possible arguments to the function.
4 -Again, if R gives rise to a descriptive function, R'x will be the class of
4 -those arguments for which the value of the function is x. Thus sin'x
consists of all numbers whose sine is x, i.e. all values of sin-lx. Again,
sin"a will be the sines of the various members of a. If a is a class of
numbers, then, by the notation of *38, 2 x"a will be the doubles of those
numbers, 3 x"a the trebles of them, and so on. To take another illustration,
let a be a pencil of lines, and let R'x be the intersection of a line x with a
given transversal. Then R"a will be the intersections of lines belonging to
the pencil with the transversal.
(2) Relations which establish a correlation between two classes are really
a particular case of relations giving rise to descriptive functions, namely the
case in which the converse relation also gives rise to a descriptive function.
In this case, the relation is "one-one," i.e. given the referent, the relatuli is
determinate, and vice versa. A relation which is to be conceived as a
correlation will generally be denoted by S or T. In such cases, we are as
a rule less interested in the particular terms x and y for which xRy, than
in classes of such terms. We generally, in such cases, have some class /3
contained in the converse domain of our relation S, and we have a class a
such that a = S"/. In this case, the relation S correlates tie members
of a and the members of f. We shall have also 3 = S"a, so that, for such
a relation, the correlation is reciprocal. Such relations are fundamental in
arithmetic, since they are used in defining what is meant by saying that two
classes (or series) have the same cardinal (or ordinal) number of terms.
(3) Relations which give rise to series will in general be denoted by P
or Q, and in propositions whose chief importance lies in their application to
series we shall also, as a rule, denote a variable relation by P or Q. When
P is used, it may be read as "precedes." Then P may be read "follows,"
--- 4 -P'x may be read "predecessors of x," P'x may be read "followers of x."
I)'P will be all members of the series generated by P except the last (if
any), (C'P will be all members of the series except the first (if any), C'P will
be all the members of the series. P"a will consist of all terms preceding
some member of a. Suppose, for example, that our series is the series of
real numbers, and that a is the class of members of an ascending series,]i X2, x3,.... Then P"a will be the segment of the real numbers
defined by this series, i.e. it will be all the predecessors of the limit of the
series. (In the event of the series x,, x2, x3,... XV,... growing without limit,
P"a will be the whole series of real numbers.)