By Leopoldo Nachbin (Eds.)

ISBN-10: 0444861785

ISBN-13: 9780444861788

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

**Sample text**

Suppose, By t h e assumption, we we g e t t h a t F(C) 5 X, c X I )= u { x : x c -z A t h a t F i s a unary o p e r a t i o n such t h a t , V A W B ( A c-B C-Z + C = n { X : F ( X ) 2 X C Z l . Since F ( Z ) 5 Z, we have now t h a t X i s such t h a t F ( X ) 5 X 5 Z. Then C C X have F(C) L F ( X ) . S i n c e we assumed t h a t F(X)-z X , f o r a l l X w i t h F(X) 5 X E Z. T h e r e f o r e F(C) 5 C. On t h e o t h e r hand, from F(C) c C 5 Z, by t h e monotony o f F , we deduce F (F(C))E F ( C )5 Z.

S)*A . ( R o S)*A = R* S*A - . - D (R o S) = S'l* D R A D ( R o S ) - l = R* D S. R CS The proof is easy. ). F i n a l l y , some o f t h e p r e v i o u s r e s u l t s a r e extended t o g e n e r a l i z e d Boolean o p e r a t i o n s . The p r o o f is l e f t t o t h e reader. 12 (iii) THEOREM SCHEMA, (nx 0"' Xn-1 Let 7 39 be a term and 4 a formula. CT : @ } ) * A C- r-X0". Xn-1 Then IT* A : $1. PROBLEMS ( x ) and ( x i ) , from ( i ) . 1. 10 2. 12. 3. 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 n t h e r e s t o f t h i s s e c t i o n , t h e l e t t e r s F, G, H, s t r i c t e d t o functions. 8 (i) (ii) = B n D F -1 F* F-'*B . F* F - ~ * B c -B . A n B = o -+ = (F*A) n B . (F-~*A) n (F-~*B) = 0 . (v) F-l*(AnB) = (F-'*A) n (F-l*B). (vi) F-l*(A%B) = (F-'*A) % (vii) (viii) (ix) 6, g, t--f DR = D S A and h. a r e r e - THEOREM (PROPERTIES OF IMAGES OF FUNCTIONS) (iii) F*(AnF-l*B) (iv) (R = S 8 5 F*A v B 3A -+ 3 C'(C' B n D F-' B n D F - ~ =F* ( X I F-~*A = F-~*B % -A C = (F-'*&). A B = F*C).

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

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