By R. Chuaqui
ISBN-10: 0080871623
ISBN-13: 9780080871622
ISBN-10: 0444861785
ISBN-13: 9780444861788
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Additional info for Axiomatic Set Theory: Impredicative Theories of Classes
Example text
Therefore y # x. Suppose ( x , y ) E R o ( - R ) - ' . ( - R -1) , i . e . ( y , z ) $ R . suchthat(x,z)ERand(z,y)€ The p r o o f Then (x,y )E of R Then t h e r e i s a z Thus, x + y . 6 REMARKS, I t i s i n t e r e s t i n g t o n o t e t h a t u s i n g some o f t h e s e i d e n t i t i e s , i t i s p o s s i b l e t o r e p l a c e e v e r y Boolean ( p r o p o s i t i o n a l c a l c u l u s ) combination o f e q u a t i o n s and i n e q u a l i t i e s by one equation. For i n s t a n c e , take (1) R = S V R ' = S' s t a r t with, l( R = S V R' = S ' ) .
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). F*A. ~-l* % B . + A nDF = B nDF . I ROLAND0 CHUAQUI 54 These p r o p e r t i e s o f images o f f u n c t i o n a r e n o t c h a r a c t e r i s t i c o f f u n c t i o n i n t h e sense t h a t t h e r e a r e o t h e r r e l a t i o n s w h i c h s a t i s f y them.
E. t o (3) - ( ( ( V x V ) o (RIS) 0 (Vx V))) Let T - = o ( V X V ) ) n ( ( V XV ) O (R’ 2 S ’ ) o 0. 5 ( x i i i ) , (3) i s f i n a l l y equivalent t o V X V ) oT o ( V x V ) = Y x V . ( 1 ) has been transformed t o t h e e q u i v a l e n t e q u a t i o n (4). 7 DEFINITION (OPERATION). 8 the 6 i d d 06 06 R ; D R-’, t h e mange 04 R ; and R. D E F I N I T I O N S (OPERATIONS). (i) AIR = R nA x V. (ii) RIB = R B . (iii) R*A = n v x Cy . : 3 x ( x € A A x R y ) } AIR i s R r e s t r i c t e d i n i t s range t o A ; R I B i s R r e s t r i c t e d i n i t s domain t o B; R*A i s t h e image o f A by R ; and R-l*B i s t h e counterimage o f 8 by R * .
Axiomatic Set Theory: Impredicative Theories of Classes by R. Chuaqui
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