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

SECTION E] PRODUCTS AND SUMS OF CLASSES OF CLASSES

329

4 -*40'682. F: U! a n 'P"i/3. 3. / C P"a
Dem.
F. *40-53. F:. Hp. D: (x): x e a: y e 3. )y. yP:
[*5-31]:()x)(:) y E /3. y. x e. yPx:
[*ll-61]: y e 3. 3Y. (3x ). x ea.yPx.
[*37-1] Dy. y e P"a:. D I. Prop
4- 4 -*40 69.: g! C'P n p'P"a.. P! P.! 'P"a
Dem.
4- - - --
-.*33-24. *24-561. D F: g! C'P n p'Pa. D. [! P.!p'P"a (1)
4 --
F. *40-62. D F: a! a. Y! 'P''a...! C'P n p'P"a (2)
F. *40-6. ) I:.a=A..: C'P p'P"a = C'P:
[*33-24] D: a! P. ). 2! C'P n p'P"a (3)
-. (2). (3).,4.83. 3 F: a! P. l p 'P"a. 3. t! C'P n p'P"a (4). (1). (4). ) F. Prop
The above propositions concerning p'R"/3 and p'R"/3 of course have
analogues for s'R"/3 and s'R"/3. But owing to *40'5, these analogues are
more simply stated as properties of R"/, and R"/3. Thus, for example,
*37'264 is the analogue of *40-67. The above propositions concerning
p'R",t and p'R",8 will be used in the theory of series, but until we reach
that stage they will seldom be referred to.
*40-7. Fl. s'a" $ = ^ {(x, y). x. ye z. z = xy}
Dem.
-. 40-11. 38-3.)
F.' sa "/3 = {(7y, y). Y e /. =? Y"a. z '7}
[*38-131] =z {(a^y,,y).y.y = y"a. xe a. z=4y}
[*13.19] =z {(Hx, y).x e a. y e/. z = y}. )- F. Prop
This proposition is of considerable importance, since it gives a compact
form for the class of all values of the function x y obtained by taking x in
the class a and y in the class /3. Thus, for example, suppose a is the class
of numbers which are multiples of 3, and /3 is the class of numbers which
are Inultiples of 5, and x x y represents the arithmetical product of x and y,
then s'a x"/3 will be the class of products of multiples of 3 and multiples
of 5, i.e. the class of multiples of 15. Again suppose a and / are both classes
of relations; then s'a ",/3 will be all relative products R S obtained by
cloosing R in the class a and S in the class /3.