errata p. 487, line 13, for" *95" read "*94."
SECTION C] SIMII..ARITY OF CLASSES
v ... *71’362. :> : Hp (4).:>. y=t= R’x. v v [*22’92.*33’44] :>. (G’R - L’Y - L’R'x)VL’R'x=G’R-L’y (5) ... (4). (5). *73’71’43. (2).:> : Hp (4). :>. (D’R - L’X) SIn (G’R -L’y) (6) ... (3). (6). :> : Hp (2). :>. (D’R - L’X) SIn (G’R - L’y) (7) t- . *51’211’22. :> t- : D’R = a. v l’X. G’R = (3 v L’Y. X r"o.) € a. y r"o.) € (3.
- >.D’R-L’X=a..G’R-L’y=(3 (8)
t- . (7) . (8) . :> : R € 1 -+ 1 . Hp (8) . :> . a. 8m f3 (9) .. . (1) . (9) . :> t- . Prop The following propositions give the proof of the Schroder-Bernstein theorem, narnely: If one class is similar to part of another, and the other is similar to part of the one, then the two classes are similar, The proof bere given is due to Zermelo*, An explanation of the following proof is given in connection with another proof in the summary of *95. v
- 73’8, .. : G’R C {3 . {:3 C D’R . IC = a (a. C ])’ R . (3 - G’R Ca.. R"a C a.) . :> .
D’R € IC. p’K C D’R Dem, . *22’42’43’44. :> : Hp. :>. D’R C ])’ R. (:3 - G’R C D’R (1) v
. *22’44 . *37’25 . :> : Hp. :> . l"D’ R D’R (2) . (1) . (2) . :> : Hp . :> . D’R € K (3)
.. . (3) . *40’12. :> . Prop
- 73’801, : Hp *73’8. :> . {3 - G’R C P’IC
Here (( Hp *7:3’8" means" the hypothesis of *73’8," Dem, t- . *20’33 . :> :. Hp . :> : a. € K . :>a. {3 - (1’ Rea. :. :> . Prop v
- 73’802, : Hp*73,g.:>. R"p’K CP’K
Dern. v
. *20’33 . :> :. Hp. :> : a. € IC . :>a .1"a. C a . (1). *40’81 . :> . Prop
- 73’81, ..: H p *73’8 . :> . P’IC € K
De-m. (1) v
.*73’8'801’802.:> : Hp.:>. p’ICCD’ R.{3-G’ RC P’IC .I"p’ICC P’IC::> . Prop
v
- 73’811, ..: Hp *73’8. :> . R"p’K C P’IC - ({3 - G’R)
Dem. v
. *37’16. :> . R"p’lC C G’R
[*22’8J C - (- G’R) [*22’81’4:3] C - «(3 - G’R)
. (1) . *73’802. :> . Prop
(1)
- Math, Annalen, vol. LXV, Heft 2, February 1908,
