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

SECTION E]

ON POWERS OF RELATIVE PRODUCTS

619

inner rectangle form (['R, and those of the outer oval form /3. Thus the
shaded portion of the figure is /38- 'R. We now define a class of classes K
by the following characteristics: a is a member of K if (1) a is contained in
D'R, (2) a contains the whole of the shaded area, (3) R"a Ca, i.e. if x is
a member of a, so is any term to which x has the relation R. Our proposition
is obtained by considering p'ic, i.e. the area common to all the members of K.
We prove (*73'81) that p'ce K, and (*73'811) that R"p'c does not contain
any of the shaded area. In the figure, R"p'c is the smaller oval. We then
prove (*73'83) that p'K consists entirely of the shaded portion and the
smaller oval. Hence 8 (the larger oval) consists of two mutually exclusive
parts, namely p'K and (PR - R"p'K, the latter being that part of the inner
rectangle which lies outside the inner oval. Assuming now that R is one-one,
\v J
p'K is similar to R"p'K; hence, adding G'R - R"p'K, it follows that 3 is
similar to ('R, and therefore to D'R.
In order to obtain hence the Schr6der-Bernstein theorem, it is only
necessary to replace R by R IS and / by (I'S, and to assume further that
S is a one-one whose domain contains (['R. Then D'R= D'(R S), and
we obtain (*73-87) (ISsm D'R, and therefore D'Ssm D'R, which was to
be proved.
In Bernstein's proof, we have the two relations R and S from the
beginning. In the left-hand part of the figure, the outer rectangle is D'R,
D'R D'S
f 3 G(R i S) Q'(SIR)
which = D'(R S), the oval is aI', and the second rectangle is (I(R S).
Thus the points of the outer but not the second rectangle form the first
generation of R I S. Within ('(R I S) we can form a third rectangle, which
will be S"R"('(R I S), i.e. ('(R S)2. The points belonging to the second
rectangle but not to the third form the second generation of R S. We can
proceed in this way to continually smaller rectangles. The points which
sooner or later are left outside some rectangle form s'gen'(R IS); those which
are common to all the rectangles form p'(I"Pot(R S). A similar analysis,