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

SjECTION B]

SC BTHE SUM OF TWO RELATIONS

353ii

*160-31. F. (1P't Q)t)~ = Pt(Q4tR)
Dem.
F. *16014-I.:)
F.P4-~(QtnR)=Pw QV RW C'P T C'6QWC'1 I C'R vC'Qt C'R (1)
F.(1). *1 60-3.:) F. Prop
*160-32. P-t-QT-R=(Pt-Q)tR Df
This definition serves merely for the avoidance of brackets.
*160-33. F:PC Q. ).P4R QtR [*:33265.-*160-1]
*160-34. F:R C S..D Q4 C Q4S [*33265. *160l]
*160-35. F: C Q.RCGS.).Pt-Q C R [*160-33034]
*160-4. F. (P v Q)t —R = (P4T-R) (Qt R)
Dent.
F.*1601.) F. (P v Q)4.R = 1)3c Q v R ' C(P 1 Q) T C('
[*33-262.*23-56] = P R Pj Q Q R P (C'P W C'Q) T C'R
[*35 41P82] = P R vi Q v -R v CIP T CIR v CIQ T CIR
[*160-1] = (P4-') R) v (Q-T-) ) F. Prop
*160-401. F. t-(Q R A) = (Pt4Q) u (Pt>)
The above two propositions state the distributive law for logical and
arithmetical addition. The three following propositions give the generalized
form of this law, when B'X replaces P v 9; these propositions are not
subsequently used but are inserted for the sake of their intrinsic interest.
*160-41. F: a! x. ). C'XtA = j'4Z"X = ~'(xtA)
Dem.
[*E160-11] r:HP))): P eX: Aly. v. xnyy. v. x e OP. y o C'RR:
[*10-42] r:(1P). P X. Py. v.(HP). PCX.xy. v.
(HP). P e X. xEG'P-y C'AR:
[*41 11.*l003 5.*41A45]: x(M'X) y.v.a!X. wLy v - e O X M. y 0' C] (1)
F-. (1). F:: Hp. DX x;6 t- -RI ). X x:S~X) Y. v. XRy. V. Xc E CBX.? e C'-R:
[*160.11] X: -'XAR) y:: DF~. Prop
*160-411. F:! x. ). P4-`X = c'P4"X [Proof as in *160-41]
*160-412. F: a! X. a jet. ). B'X zI =a!
Dem.
F. *1 601.-411 D F M B~~B'/L '(M') t- "/.k(1
t-. 160-41 -:~ih.P~.Bh~'u =B~~~~ (2)
[*42-12] = 's'x cc/.: ) F. Prop
R.& W. II. 23