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

*70. RELATIONS WHOSE CLASSES OF REFERENTS AND OF RELATA
BELONG TO GIVEN CLASSES.
Summtary of *70.
If a and /3 are two given classes of classes, a relation R is said to belong
--
to the class a -,/ if R'y e a whenever y e (['R, anld R 'xe /3 whenever x e D' R.
If only one of tliesc conditions is to be imposed, this result is secured by
replacing the class involved in the other condition by " Cis," since " R'Y e Cs "
4 -always holds, and so does "R'xeCls," and therefore neither imposes anly
limitation on R. In the most important cases, a and 8/ are either both
cardinal numbers, or one is a cardinal number while the other is Cls.
In virtue of *37'702'703, the conditions above mentioned as imposed
upon R by membership of a-,8/ are equivalent to
-4 -R "(PIR C a. R "D'R C 3.
This form is used in the definition (*70'01).
The propositions of the present number are hardly ever used except in
*71, where a and /3 are both replaced by 1 or CIs. The most useful
propositions are
44 -*701. F: Rae /-3. -. R"a('R C a.R"D'R C3,
(This merely embodies the definition.)
4 —
*7013.. H:. Rea-3.: (y).R'y eau 'A: (x).R'xe/3 v 'A
*70-22. F. /3 - a = Cnv"(a —,/3)
*70-4. F. a -Cls = R (RR"'R C a)
*70-41. F. Cls ---> = R (R"D'CR C 3)
*70'42. F. a — / = (a - Cls) n (Cls — 3)
*70'54. F: (I'R n (I'S= A. R, Se a -Cls.. R W Se a - Cls
with similar propositions for Cls -, / and a - /3.
*70-62. F: R a - Cls.. R y e a —Cls
with a similar proposition for Cls - /3.