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

SECTION B] ARITHMETICAL SUM OF TWO CLASSES AND TWO CARDINALS 7

77

xy.e/a. R = (A (A)(')
[*38-13-131. (*11.001)]
*110-101. 1-.(tx)4I (A A/3) +(A A a)4I (t'y)
Dern.
F. *55156. ) F. t)~x) 4 (A r% 3 ''.'A )4(')=t( a).(1
[*512:3] ) F. tl~ +# t'(A A a) (2)
F.(1). (2). ) F. I)'"(t'w) 4 (A A /3) + 1)'(A A a4 ty). D F. Prop
Dern.
F.*110-101. )F xea. A= 4(A A /)'t'x.y~,E 3.8S=(A A a) 4't'y. ).R S
[*37-67] F: RIe I(A n/3)"t"a.S8e (A na)4I "f/3. D. R#+S(1
F. (1). *24-37. D F. Prop
*110-12. F.4(A/)"a ma.(Aa)4""3m/3[*7341-61-611]
*110-11-12 give the juistification for the use of a + ~3 in defining arithmietical addition, since they show that a ~ /3 consists- of two mutually
exclusive parts wvhich -are respectively similar to a and /3.
*110-13. F:-ysina.8sm/3,.,yAS=A.:).,yv~sm~(a~/3)
Dern.
F.*110-12.DF: Hp.D.,ysm 4 (A A/3)"t"a.3smi(Ana) 4"t"/3 (1)
F.(l). *110-11. *73-71.)D F. Prop
*110-14. F:aA/3=A.D. a v/3sm(a~/3) [*.110-13.*73-3]
Thus whenever a and /3 are mutually exclusive, their logical suim may
replace their arithmetical sumir in defining the suit of their cardinal numbers.
*110-15. F:,ysma.8sm/,3.D.,y~8st-a~/3
Dent.
F.*110-12. )F: Hp.).4(AA8)",tllysma.(AA-y)4 't'8sm/3 (1)
F. *110-11 D F. 4 (AA r)"t'"ryA (A ny)4 "t"36= A (2)
F.(I). (2) *110-13.)D
F: lip. D. 4 (A A S)"tcy u(A A y) 4 "t"8 sm a ~ /3: D F. Prop
*110-151.
Dem.
F. *73-71. D F:. lip):D
F. *72-411. *37-25-22.*73-22.)D
F: Sc I-*I. D'S = ~ C1S a v/3.aAr 3 =A.)D.
Stca A% S"/3 = A.~ = S66a v S"/j3. S"a sm a. S"/38 smn/3.
F. (2). *10-1123-35. *73-1.)
F:.Hp.):~smn(av/3).).(Hy,8).,ysma.3sm/3.,yA3=A.4=,yv3 (3)
F.(I). (3).)DF. Prop