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

464

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

*72-92. F: Re - Cls. SC R. ). S= R r 'S
This proposition shows that every relation contained in a one-many
relation is obtainable by a limitation of the converse domain. Thus e.g.
every relation contained in that of father to son can be specified by specifying
the class of sons who are to be its converse domain; for then all the fathers
of these sons must be included to provide referents. But if we take the
relation of parent and child, which is not one-many or many-one, a contained
relation is not determinate even when both its domain and its converse
domain are given; for the relation may relate some of the children in any
one family to the father and some to the mother, and so long as all the
children and both parents are each related to some one by the relation, the
domain and converse domain remain unchanged by permutations within the
family.,72-1. F.Ael->l
Dem.
F.,25-105. -. (xAz. yAz).
[*2'21] ) -: xAz. yAz.. x= y:
[*11'11.*71'17];) F. A e 1 - Cls (1)
Similarly. Ae Cls -- 1 (2)
F. (1).(2). *71-103. D). Prop,72-11. F. Cnv e 1 -- 1
Dem.
F-. *31-13. *71166. D. Cnve l- Cls (1)
F. (1). *71-54. 31'32-12. ). Prop
*72-12. F. R, R e 1 - Cls [*3212-121. *71-166]
*72-121. F. sg, gs e I - 1
Dem.
F. *32-22221. *71166. D F.sg, gs 1 -- Cls (1)
F. (1). *32-14'15-21'211. *7154. F. Prop
*72-13. F. De -Cls [*33'12. *71-166]
*72-131. F. I e 1 - Cls [*33-121. *71166]
*72'132. F: C e 1 - Cls [*33-122. *71166]
*72'14. F. x $,?x e 1 - Cls [*38-12. 71-166]
This proposition applies to a great many of the relations we have to deal
with, for example 1 P, P r, Pt, P, IP, x J,, x, etc.
*7215. F. Pee 1 -- Cls [*37'111. *71166]
In *72'16 below, p has the meaning defined in *40'01, and does not
represent a variable proposition. Similarly s in *72'161 has the meaning
defined in *40-02.