ENGR 3861 Digital Logic Boolean Algebra. Fall 2007

Transcription

1 ENGR 386 Digitl Logi Boole Alger Fll 007

2 Boole Alger A two vlued lgeri system Iveted y George Boole i 854 Very similr to the lger tht you lredy kow Sme opertios ivolved dditio sutrtio multiplitio Repled y their Boole equivlets Sho dpted Boole lger to work with rely iruits Relys re either o or off Digitl iruits e lso e thought of i similr wy Either oe of two sttes low voltge (or off 0 volts) stte or i high voltge (or o 5 volts) stte Therefore we use Boole lger to desrie moder digitl iruits

3 Some si defiitios to strt Axiom A strtig ssumptio from whih other sttemets or theorems re logilly defied They ot e derived y priiples of idutio d ot e demostrted y forml proofs Not eessrily ovious or self-evidet ut re epted s truth to use to yield further results Theorem A sttemet tht e proved usig previously greed upo ssumptios (Axioms) Are derivle usig fixed set of dedutio rules 3

4 Boole Alger - Defied Cosists of: A. A set of elemets B B. Three opertors ' d C. A set of 0 xioms 4

5 The 0 Axioms if if 0 the the 0 Bsilly stte tht either e or 0 is referred to s -prime or NOT Defie the AND or Boole multiplitio opertor Defie the OR or Boole dditio opertor 5

6 Truth Tles A Truth Tle is used i Boole lger to ompute the futiol vlues of Boole expressio for eh omitio the futio s iput vriles X Y Z F(XYZ) F(000) 0 0 F(00) 0 0 F(00) 0 F(0) 0 0 F(00) 0 F(0) 0 F(0) F() 6

7 Sigle Vrile Theorems idetities ull ivolutio elemets idempotey omplemets 0 To prove y of these theorems we just write out truth tle sed o xioms Ex: += + 0 7

8 Two d Three Vrile Theorems 8 omiig erig distriutivity ssoitivity ommuttivity ov

9 Theorems with -vriles 9 Expsio Theorem Sho s F F F F F F s DeMorg Geerlized F s Theorem DeMorg idempotey geerlized ' 0 ) ( 0 ) ( 4. ' ) ( ) F( 3. '..

10 So wht do we do with ll these? Proofs Use them to simplify lrge Boole expressios You will see why simplifitio is useful i the ext lss or two Exmples See ord otes 0

11 A use of the distriutivity theorem C e used to oti sum-of-produts (SOP) from produt-of-sums (POS) d vie vers Exmple Uses the theorem: Strt y lettig: This gives: z x y x z w y w z v y v z y x w v x w v z y z y x w v z x w v y x w v

12 A use of distriutivity otiued The tke two terms: v w x y d v w x z Ad use the sme theorem gi o eh to get fil result The importe of eig le to overt etwee the two forms will eome ppret lter i the ourse

13 Dulity Notie tht ll xioms were i pirs Result of the priiple of Dulity Ay theorem or idetity i oole lger remis true if 0 d re swpped d d + re swpped throughout Importt euse it doules the usefuless of ythig we ler here i this ourse Note tht the dul is ot the sme s pplyig DeMorg s Theorem Also e reful i pplyig the dulity rule! 3

14 Dulity Cutio! For exmple: x x y x x x y y But this is wrog!!! Wht hppeed? x x x y dulity from xiom Mke sure to prethesize expressio fully efore pplyig the priiple of dulity x x x y x x y x y dulity 4

15 From lger to iruits Eh of the logil opertios hs equivlet iruit symol NOT AND 5

16 Bsi Logi Symols - otiued OR 6

17 Some Simple Exmples Show the iruit equivlet of the followig Boole futios: F=(+)+ F=(+)d + F=(xyz+x )y Determie the lgeri expressio for the followig iruit: 7

18 More th iputs F-i desries the umer of iputs to logi gte Eg: F-i = 4 F-i = F-out desries the umer of gte iputs drive y gte output 8

19 More logi opertors NAND/NOR Result of DeMorg s Theorem We kow tht: x y x y Grphilly we show this s: 9

20 NAND/NOR Cotiued Could lso write DeMorg s s: ( x y) x y Whih we show grphilly s: 0

21 Uses of NAND/NOR Gtes NAND d NOR gtes e used s uiversl gtes i.e. implemet ll other si gtes usig just NANDs or NORs Clled futiol ompleteess Eg: As exerise figure out the NOR sed equivlets!

22 Exmples: Implemet the followig iruits usig oly -iput NAND gtes

23 Why use NANDs? Esiest to mufture of ll the gtes Fster th other gte implemettios Covertig to NANDs result i more ompt iruit reliztio 3