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

SECTION D]

DOMAINS AND FIELDS OF RELATIONS

261

kinds. Consider first the sort of relation that gives rise to a descriptive
function R'y. For this we require that R'y should exist whenever there is
anything having the relation R to y, i.e. that there should never be more
than one term having the relation R to a given term y. In this case, the
values of y for which R'y exists will constitute the "converse domain " of R,
i.e. (I'R, and the values which R'y assumes for various values of y will
constitute the "domain" of R, i.e. D'R. Thus the converse domain is the
class of possible arguments for the descriptive function R'y, and the domain
is the class of all values of the function. Thus, for example, if R is the
relation of the square of an integer y to y, then R'y= the square of y,
provided y is an integer. In this case, (I'R is the class of integers, and D'R
is the class of perfect squares. Or again, suppose R is the relation of wife
to husband; then R'y= the wife of y, I'R = married men, D'R = married
women. In such cases, the field usually has little importance; and if the
values of the function R'y are not of the same type as its arguments, i.e. if
the relation R is not homogeneous, the field is meaningless. Thus, for example,
if R is a homogeneous relation, R and R are not homogeneous, and therefore
— ~ 4 -" C'R" and " C'R" are meaningless.
Let us next suppose that R is the sort of relation that generates a series,
say the relation of less to greater among integers. Then D'R = all integers
that are less than some other integer = all integers, I'R = all integers that
are greater than some other integer =all integers except 0. In this case,
C'R=all integers that are either greater or less than some other integer
= all integers. Generally, if R generates a series, D'R = all members of the
series except the last (if any), a('R = all members of the series except the first
(if any), and C'R = all members of the series. In this case, " xFR" expresses
the fact that x is a member of the series. Thus when R generates a series,
C'R becomes important, and the relation F is likely to be useful.
We shall have occasion to deal with many relations having some of the
properties of series, and wi'th many propositions which, though only important
in connection with serial relations, hold much more generally. In such cases,
the field of a relation is likely to be important. Thus in the section on
Induction (Part II, Section E), where we are preparing the way for the
construction of serial relations by means of a certain kind of non-serial
relation, and throughout relation-arithmetic (Part IV), the fields of relations
will occur constantly. But in the earlier parts of the work, it is chiefly
domains and converse domains that occur.
Among the more important properties of domains, converse domains and
fields, which are proved in the present number, are the following.
We have always E! D'R, E! (I'R, E! C'R (*3312'121'122). (The last of
these, however, is only significant when R is homogeneous.)