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

SECTION B]

DERIVATIVES

701

If P is a series, the derivative of a class a consists of those members x of
(IP which are such that members of a exist in every interval which ends
in x, i.e.
4- --
*216'13.:: P e Ser.:. e p'a. _-: x e (P: yPx. y. [! a n P'y n Px
We have
*216-2. F. Gp'C'P = D'ltp- B'P
*216 3.: a e dense'P.. a - min'a C 8p'a
*216-32. -: a e closed'P.. Cl ex'(a n C'P)C (I'limaxp. 3e'a C a
We prove (*216*4 —412) that the properties of a with respect to P, as
regards being dense, closed, or perfect, belong to S"a with respect to Q if S
is a correlator of P with Q.
We next consider the relation of a in P to P"a in 3'P (*216'5 —56).
The point of these propositions is that s'P is Dedekindian, so that a class is
closed in s'P if it contains its first derivative. (It is usual to define a class
as closed whenever it contains its first derivative; but this involves the tacit
assumption that the series P is Dedekindian. If P is the series of real
numbers, this assumption is of course verified.) We prove (*216'52) that
the derivative of P"a in s'P is P"'(C1 ex'a - ('maxp), i.e. is the class of
segments defined by such existent sub-classes of a as have no maximum; we
show that a is dense, closed, or perfect in P according as P"a is dense, closed,
or perfect in s'P (*216'53'54'56), and that a and P"a are closed if P"oa
contains its first derivative (*216'54).
We end with various propositions on V'P (*216'6-'621), of which the
chief is
*216'611. F: P e Ser. 3! V'P. ). C'V'P = C'P - ('P = pCc'P v B'P
This subject will be resumed in connection with well-ordered series
in *264.
*216 01. Sp'a = ltp"Cl ex'(a n C'P) Df
*216'02. dense'P = a (a - minp'a C,p'a) Df
*216'03. closed'P = a {Cl ex'(a n C'P) C I'limaxp. Ap'a C a} Df
*216-04. perf'P = dense'P n closed'P Df
*216'05. V'P =P D'ltp Df
*216.1. F: x e pla. -. (3/8). /3 C a n CP. a! /. x ltp /3 [(*216-01)]