By Paul R. Halmos

From the reports: "...He (the writer) makes use of the language and notation of standard casual arithmetic to kingdom the elemental set-theoretic proof which a starting scholar of complex arithmetic must know...Because of the casual approach to presentation, the publication is eminently suited to use as a textbook or for self-study. The reader should still derive from this quantity a greatest of realizing of the theorems of set thought and in their uncomplicated value within the research of mathematics." - "Philosophy and Phenomenological Research".

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We a l s o assume i n t h i s Chapter t h a t R, S, T a r e r e l a t i o n s . 1 (i) R0R-l cID(ii) R O R - ~ 510- R O R - ~ (v) R OR-^ LIDc -D I (vi) RoR-'cID- ( ~ v Ro) n R = 0. WSWT(SnT)oR = (SoR) n ( T O R ) . ( i i i ) R o R - lcID(iv) ( R o R - 1) n D v = 0 . - W S W T ( S ~ T ) =~ R ( s ~ R 2 ), (TOR). wx WY R - ~ * ( xn Y ) = ( R - ~ * x )n ( R - ~ * Y ) . V X W Y (R* X) f- Y = R*(XnR-'*Y). I t i s c l e a r t h a t o f o u r c o n s t a n t r e l a t i o n s , I D and 0 a r e f u n c t i o n s , w h i l e V x V , E L , I N , and Dv a r e n o t .

T h e r e f o r e , F-l*A n F-l*B n D F = 0 . F-l*B C - D F . Thus, F-l*A n F-l*B = 0. B u t F-'*A PROOFOF ( v ) . By ( i v ) , we have ( F - 1 * ( A n 8 ) ) n ( F - 1 * ( 8 n J A ) ) = ( F - l * ( A n B ) n (F-'*(A%E)) ( v ) , we o b t a i n , F-l*A = (F-l*(A%B)) = 0 = From 2 . 2 . 1 . 1 0 n (F-l*(B\A)). (F-'*(AnB)) U (F-l*(A-E)) a n d F-'*B = T h e r e f o r e (Fql*A) n (F-'*E)=((( F - l * ( A n E ) ) U = (F-l*(AnB)) u (F-l*(B%A)). (F-'*(A = * 8 ) ) )) n (( (F-l*(A PROOF OF ( v i ) . n8 ) ) U (F-'*( E % A ) ) ) = F-l*(A n B ) By 2 .

Show t h a t : ( i ) R o ( S n T ) = ( R o S ) n ( R o T ) i s not t r u e i n general, ( i i ) (R = 0 V S = 0 ) - R o ( V x V ) o S = 0, ( i i i ) R n S n T C- R o S - ' o T . ( i v ) R* (AnB) = (R*A) n (R*B) i s not true i n general. 4. - ) , : 3 z(q = ( x , z ) ) l , . - A. e. 2 RELATIONS A S SUPERCLASSES, A c o l l e c t i o n , o r any n o t i o n , i s defined by a formula. In general, c o l l e c t i o n s a r e not c l a s s e s . However, t h e r e a r e some c o l l e c t i o n s which can be represented by r e l a t i o n s which a r e themselves c l a s s e s .

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