# Axiomatic Set Theory: Theory Impredicative Theories of by Leopoldo Nachbin (Eds.) By Leopoldo Nachbin (Eds.)

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Extra resources for Axiomatic Set Theory: Theory Impredicative Theories of Classes

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V i i ) R o (S U T ) = ( R o S ) ( v i i i ) (S U T) o R = (S U (R o T). o R ) u ( T o R). - R o T A S o R c- T (ix) S E T + R o S c ( x ) R o (S n T ) 2 (R o S) n (R o T). ( x i ) (S n T ) o R C ( S o R ) n (T o R ) . o R (xii) R o 0 = 0 = 0. - o R. ( x i i i ) ( V X V ) O R O( V X V ) = V X V - R + O . (xiv) (V x V )o R o (V x V ) = 0 ( x v ) ( R o S)-' = S-l o R-'. R = 0. ( x v i ) R n ( - R ) - ~c -D Y ( x v i i ) R o ( - R)-' C - D ~ PROOF, The proof o f these statements i s q u i t e easy.

For i n s t a n c e , we have when F i s a unary o p e r a t i o n , { ( F ( x ) , x ) : F(x) E V ) . (F(x): F(x)E V ) = X We can i n t r o d u c e f u n c t i o n s f o r a l l t h e o p e r a t i o n s a l r e a d y d e f i n e d . For example, f o r t h e image o p e r a t i o n , R* = ( R * x : x E v) . I n t h i s case and o t h e r s , t h e same symbol w i l l be used f o r t h e operat i o n and t h e corresponding f u n c t i o n . 4 can be g i v e n by, F*A = { F ' x : x E A } , and F-l*A have e a s i l y , tl x ( x t h e image o f a f u n c t i o n F, = { x : F'xEA}.

On t h e o t h e r hand A ( ? ) i s t h e no;tion t h a t says t h a t f 12 a 6unOtiun w a h domain 8 m d tange i n c h d e d in A. W i t h t h i s n o t i o n , we can express t h a t ? i s a f u n c t i o n by D F V ( F ) . T h i s w i l l o f t e n be used. 11 DEFINITION SCHEMA (GENERALIZED CARTESIAN PRODUCT). Let be a term and \$ a formula. Then n ( 7 :q,) = Ed : 6 & x V A ~ ; O ~ - ' C J D A DI x~: = \$1 A Vx(\$ + 6kE r)). 12 For instance, we have F, by F ( 0 ) = A xEA A "6 = nX (6'~:x DEFINITION, gEBl.