Boolean Algebra cont. The digital abstraction

Transcription

1 Boolen Alger ont The igitl strtion

2 Theorem: Asorption Lw For every pir o elements B. + =. ( + ) = Proo: () Ientity Distriutivity Commuttivity Theorem: For ny B + = Ientity () ulity.

3 Theorem: Assoitive Lw In Boolen lger eh o the inry opertions ( + ) n ( ) is ssoitive. Tht is or every B. + ( + ) = ( + ) +. ( ) = ( )

4 Proo: () Let A Distriutivity A Commuttivity Distriutivity Distriutivity Iempotent Lw Asorption Lw Asorption Lw

5 A Commuttivity Distriutivity Distriutivity Iempotent Lw Asorption Lw Commuttivity Asorption Lw

6 Putting it ll together: A Sme trnsitions eore +

7 A A () Dulity Also

8 Theorem : DeMorgn s Lw For every pir o elements B. ( + ) =. ( ) = + Proo: () We irst prove tht (+) is the omplement o. Thus (+) = By the einition o the omplement n its uniqueness it suies to show: (i) (+)+( ) = n (ii) (+)( ) =. () Dulity ( ) = +

9 Distriutivity Commuttivity Assoitivity n re the omplements o n respetively Theorem: For ny B + = Iempotent Lw

10 Commuttivity Distriutivity Commuttivity Assoitivity Commuttivity n re the omplements o n respetively Theorem: For ny B = Iempotent Lw

11 Alger o Sets Consier set S. B = ll the susets o S (enote y P(S)). plus set-union times set-intersetion M P S Aitive ientity element empty set Ø Multiplitive ientity element the set S. Complement o B: S \

12 Theorem: The lger o sets is Boolen lger. Proo: By stisying the ioms o Boolen lger: B is set o t lest two elements For every non empty set S: S PS B. Closure o ( ) n ( ) over B (untions B B B ). S. P(S) y einition P(S) y einition S n S n P( S) P( S) y einition y einition

13 A. Cummuttivity o ( ) n ( ). or : or : An element lies in the union preisely when it lies in one o the two sets n. qully n element lies in the union preisely when it lies in one o the two sets n. Hene n : n :

14 A. Distriutivity o ( ) n ( ).. Let n or I We hve n. Hene I We hve n. Hene or

15 This n e onute in the sme mnner s. We present n lterntive wy: Deinition o intersetion n Also einition o intersetion einition o union Similrly * **

16 Tking (*) n (**) we get Distriutivity o union over intersetion n e onute in the sme mnner.

17 A3. istene o itive n multiplitive ientity element. S. S S S. itive ientity - multiplitive ientity - S A4. istene o the omplement. S B \. S S S B \ \. S S B \ \. Alger o sets is Boolen lger. All ioms re stisie

18 Boolen epression - Reursive einition: se: B epressions. reursion step: Let n e Boolen epressions. Then ( + ) ( ) Dul trnsormtion - Reursive einition: Dul: epressions epressions se: B\{} reursion step: Let n e Boolen epressions. Then [ul( )] ( + ) [ ul( ) ul( ) ] ( ) [ ul( ) + ul( ) ]

19 Let e the ul o untion ( n ) Lemm: In swithing lger = ( n ) Proo: Let ( n ) e Boolen epression. We show tht pplying the omplement on the whole epression together with repling eh vrile y it s omplement yiels the ul trnsormtion einition. Inution sis: epressions. n n

20 Inution hypothesis: Lemm hols or Boolen epressions: n. Tht is: n n n n n n Inution step: show tht it is true or ( + ) ( ) n n hypothesis inution n n Lw Morgn De' n n I then n

21 I then hypothesis De' Morgn Lw I then inution inution hypothesis

22 Deinition: A untion is lle sel-ul i = Lemm: For ny untion n ny two-vlue vrile A the untion g = A + A is sel-ul. Proo: (hols or ny Boolen lger) ul g ula A Dul einition Distriutivity Commuttivity ul A ul A ula ul ula ul A A A AA A A A

23 Distriutivity Commuttivity A A A AA AA A A A A A is the omplement o A A A Ientity A A Commuttivity A A Notie tht the ove epression hs the orm: + + where =A = =.

24 We now prove stronger lim: B. Ientity is the omplement o Distriutivity Commuttivity Commuttivity Distriutivity Theorem: For ny B + = Ientity

25 ul g ula A A A A A For emple: v g v v v sel-ul

26 sier proo () or swithing lger only: (using ul properties) ul g ula A A A Swithing lger n n OR Ientity A A A A

27 sier proo () or swithing lger only: (se nlysis) ul g ula A A A A = = Ientity Commuttivity Theorem: For ny B = Ientity ul( g) Asorption Lw g

28 A = ul( g) g

29 mple o trnser untion or n inverter : stritly eresing in. stritly inresing in onve in the intervl. is onve in the intervl monotone eresing

30 - - - ontinuous!.!. slope = - slope = -

31 slope = - high out slope = - low out low in high in true only i: high out high in low in low out

32 BUT this is not lwys the se. For emple: slope = - high out slope = - low out low in high in high in high out Moreover in this emple it n e prove tht no threshol vlues eist whih re onsistent with einition 3 rom leture notes.

33 Using the ssumption: there eists suh tht point : () = strt with : high out high in low in low out y y slope < -

34 set : high in high in () = low out high in y y low in low in high out low in slope < -

35 slope = - () = high in low out low in high out high out slope = - high out high in low in low out low out true i: low in high in min slope < - set min