Boolean-Algebra

# Boolean-Algebra - 1 B O O L E A N A L G E B R A S • I m p o r t a n t c l a s s o f a l g e b r a s e x t e n s i v e l y u s e d f o r m a n y p

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Unformatted text preview: 1 B O O L E A N A L G E B R A S • I m p o r t a n t c l a s s o f a l g e b r a s e x t e n s i v e l y u s e d f o r m a n y p u r p o s e s • B a s i s o f t h e s w i t c h i n g a l g e b r a ( S A ) f o r f o r m a l t r e a t m e n t o f s w i t c h i n g c i r- c u i t s : – T r a n s f o r m a t i o n o f s w i t c h i n g e x p r e s s i o n s – I d e n t i t i e s f r o m B A e n a b l e g r a p h i c a l a n d t a b u l a r t e c h n i q u e s f o r m i n i m i z a- t i o n o f s w i t c h i n g e x p r e s s i o n s I n t r o d u c t i o n t o D i g i t a l S y s t e m s A p p e n d i x A – B o o l e a n A l g e b r a s 2 D E F I N I T I O N O F B O O L E A N A L G E B R A A B o o l e a n a l g e b r a i s a t u p l e { B , , · } : • B i s a s e t o f e l e m e n t s ; • a n d · a r e b i n a r y o p e r a t i o n s a p p l i e d o v e r t h e e l e m e n t s o f B , s a t i s f y i n g t h e f o l l o w i n g p o s t u l a t e s : P 1 : I f a , b ∈ B , t h e n ( i ) a b = b a ( i i ) a · b = b · a T h a t i s , a n d · a r e c o m m u t a t i v e . P 2 : I f a , b , c ∈ B , t h e n ( i ) a ( b · c ) = ( a b ) · ( a c ) ( i i ) a · ( b c ) = ( a · b ) ( a · c ) I n t r o d u c t i o n t o D i g i t a l S y s t e m s A p p e n d i x A – B o o l e a n A l g e b r a s 3 P 3 : T h e s e t B h a s t w o d i s t i n c t i d e n t i t y e l e m e n t s , d e n o t e d a s a n d 1 , s u c h t h a t f o r e v e r y e l e m e n t i n B ( i ) a = a = a ( i i ) 1 · a = a · 1 = a T h e e l e m e n t s a n d 1 a r e c a l l e d t h e a d d i t i v e i d e n t i t y e l e m e n t a n d t h e m u l t i p l i c a t i v e i d e n t i t y e l e m e n t , r e s p e c t i v e l y . P 4 : F o r e v e r y e l e m e n t a ∈ B t h e r e e x i s t s a n e l e m e n t a , c a l l e d t h e c o m p l e m e n t o f a , s u c h t h a t ( i ) a a = 1 ( i i ) a · a = I n t r o d u c t i o n t o D i g i t a l S y s t e m s A p p e n d i x A – B o o l e a n A l g e b r a s 4 C O M M E N T S • S y m b o l s a n d · a r e n o t t h e a r i t h m e t i c a d d i t i o n a n d m u l t i p l i c a t i o n s y m b o l s F o r c o n v e n i e n c e a n d · c a l l e d “ p l u s ” a n d “ t i m e s ” E...
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## This note was uploaded on 04/17/2008 for the course CS 151A taught by Professor Miloseragovich during the Fall '07 term at UCLA.

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Boolean-Algebra - 1 B O O L E A N A L G E B R A S • I m p o r t a n t c l a s s o f a l g e b r a s e x t e n s i v e l y u s e d f o r m a n y p

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