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

SECTION D] CLASSES OF MUTUALLY EXCLUSIVE CLASSES

545

*84'342. F: K e Cls2 excl. R e ea'K. a e K.. 'R'a = a D'R
[*8115. *84-14. *62-2]
*84-35. F:. K 6 Cs exexcl. D: REe,K.-.R e 1 -. RCe. R = I e P K
Demn.
F.*8413.: Hp.. A e.
[*62-42] ). P' K =K (1)
F. (1). *71'103. *80-14. D
F:.Hp.D: R e —. R e. C(I'R = ('e r Ic. D. R e ec' (2)
F. (1). *80-14. D F:. Hp. D: R e ca'. D..CR = WE I K (3)
F. (3). *80291.*843. )
F:.Hp. ): REESK. ).lR -l. It C 'R.R=a rK (4)
F. (2). (4). D F. Prop
*84-37. F: K e Cls2 excl.! 6e'jK. D. K e Cls ex2 excl [*831. *84-13]
*84-4.:. Ke CIs2 excl. R,S e eaK. D:D'R -DS._. R = S [*812. *84-14]
*84'41. F: e Cls2 excl. D. D ' ea'l e 1 —+ 1.D"eK smn 6A' [*81'21. *84-14]
This is an important proposition, since it shows that, when K is a
Cls2 excl, the number of classes that can be selected from K is the product of
the numbers of the various classes that ate members of K.
*84-411.. a e K. Da./ a e 1 C s'K: D. e D"E [*81'212. *62-2'3]
*84-412. F: K e Cls2 excl. D. D" = {a e.. t, n a e 1: t:C s'K}
[*81-22. *84-14. *62-2'3]
This proposition gives what might be taken as the definition of the class
of selected classes, namely
^ {a. * lt A^ a E 1e: /LC s'Kc}.
We might, starting with this as our definition, deal with the class of
selected classes without first considering selective relations. The disadvantages
of this method would be, first, that it requires that K should be a CIs2 excl if
it is to give the results desired in arithmetic; secondly, that it is much more
cumbrous technically than the method which proceeds by selective relations;
thirdly, that it does not enable us to deal with selection from a class of
classes as a particular case of selection from a relation (nanmely from e c K),
and therefore does not yield theorems of such generality as those obtained
by the method adopted above.
*84-42. F: K e Cls2 excl. a e K. tx e D"ea'K.. - a e D"e'(K - t'a)
[*8124. *84-14. *62-2]
*84-421. F: a e K. x e a./~ e D"e6'(K - 'a). V, V t'x e D"eA
[*81-25]

R. &W.

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