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

SECTION D]

LIMITED DOMAINS AND CONVERSE DOMAINS

279

Relations with limited domains are not nearly so much used as relations
with limited converse domains. Relations with limited converse domains
play a great part in arithmetic, especially in establishing the formal laws.
What is wanted in such cases is a one-one relation correlating two classes or
two series. That is, we want a relation such that not only does R'y exist
whenever ye ('R, but also R'x exists whenever xeD'R. The kind of
relation which is most frequently found to effect such a correlation is some
such relation as D or ( or C, or some other constant relation for which we
always have E! R'y, with its converse domain so limited that, subject to
the limitation, only one value of y gives any given value of R'y. Thus for
example let X be a class of relations no two of which have the same domain;
then D [ X will give a one-one correlation of these relations with their
domains: if R, SeX, we shall have
D'R = D'S. D. R = S.
We shall also have D'R =(D X)'R and D'S = (D X)'S. Moreover the
converse domain of D X is X, and the domain of D X is the class of
domains of members of X. Thus D HX gives a one-one correlation of X with
the domains of members of X. It is chiefly in such ways that relations with
limited converse domains are useful.
For purposes of reference, a great many propositions are given in the
present number, but the propositions that will be used frequently are comparatively few. Among these are the following:
*35-21. F.alRr,= =(a R) /3= t(Rr3)
*35'31. F. (R a)3 = R r(a n 8)
*35354. F.(Rra)lS= R a1S
I.e. in a relative product it makes no difference whether we limit the
converse domain of the first factor, or the domain of the second.
*35412. F.R ' (f, u ')= R P 3 R rP3'
*35.452. F: aC'R C.. R P = R
*3548. F:('PCa.D.P|(a1R)=P R
*35-52. F. Cnv'(R rP ) = 3 1 R
*35.61.. D'(a 1 R) = a n D'R
*35'64. F. ('(R 83) = 38 n r'R
*35-65. F: 3 C (SR.. d'(Rr P8)=/3
The hypothesis 3 C (I'R is fulfilled in the great majority of cases in
which we have occasion to use R /,3.
*35'66. F: (I'R C/..R r/3 R
~35'7. F: c {(R r f)'Y. - * y e 3. ( R'y)