Boolean Algebra. ECE 152A Winter 2012
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1 Boolen Algebr ECE 52A Wnter 22
2 Redng Assgnent Brown nd Vrnesc 2 Introducton to Logc Crcuts 2.5 Boolen Algebr 2.5. The Venn Dgr Notton nd Ternology Precedence of Opertons 2.6 Synthess Usng AND, OR nd NOT Gtes 2.6. Su-of-Products nd Product of Sus Fors Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2
3 Redng Assgnent Brown nd Vrnesc (cont) 2 Introducton to Logc Crcuts (cont) 2.7 NAND nd NOR Logc Networks 2.8 Desgn Exples 2.8. Three-Wy Lght Control Multplexer Crcut Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3
4 Redng Assgnent Roth 2 Boolen Algebr 2.3 Boolen Expressons nd Truth Tbles 2.4 Bsc Theores 2.5 Couttve, Assoctve, nd Dstrbutve Lws 2.6 Splfcton Theores 2.7 Multplyng Out nd Fctorng 2.8 DeMorgn s Lws Jnury, 22 ECE 52A - Dgtl Desgn Prncples 4
5 Redng Assgnent Roth (cont) 3 Boolen Algebr (Contnued) 3. Multplyng Out nd Fctorng Expressons 3.2 Exclusve-OR nd Equvlence Operton 3.3 The Consensus Theore 3.4 Algebrc Splfcton of Swtchng Expressons Jnury, 22 ECE 52A - Dgtl Desgn Prncples 5
6 Redng Assgnent Roth (cont) 4 Applctons of Boolen Algebr Mnter nd Mxter Expressons 4.3 Mnter nd Mxter Expnsons 7 Mult-Level Gte Crcuts NAND nd NOR Gtes 7.2 NAND nd NOR Gtes 7.3 Desgn of Two-Level Crcuts Usng NAND nd NOR Gtes 7.5 Crcut Converson Usng Alterntve Gte Sybols Jnury, 22 ECE 52A - Dgtl Desgn Prncples 6
7 Boolen Algebr Axos of Boolen Algebr Axos generlly presented wthout proof = + = = + = = = + = + = f X =, then X = f X =, then X = Jnury, 22 ECE 52A - Dgtl Desgn Prncples 7
8 Boolen Algebr The Prncple of Dulty fro Zv Kohv, Swtchng nd Fnte Autot Theory We observe tht ll the precedng propertes re grouped n prs. Wthn ech pr one stteent cn be obtned fro the other by nterchngng the OR nd AND opertons nd replcng the constnts nd by nd respectvely. Any two stteents or theores whch hve ths property re clled dul, nd ths qulty of dulty whch chrcterzes swtchng lgebr s known s the prncple of dulty. It stes fro the syetry of the postultes nd defntons of swtchng lgebr wth respect to the two opertons nd the two constnts. The plcton of the concept of dulty s tht t s necessry to prove only one of ech pr of stteents, nd ts dul s henceforth proved. Jnury, 22 ECE 52A - Dgtl Desgn Prncples 8
9 Boolen Algebr Sngle-Vrble Theores Theores cn be proven wth truth tbles Truth tble proof.k.., Perfect Inducton X = X + = X = X X + = X X X = X X + X = X X X = X + X = (X ) = X Jnury, 22 ECE 52A - Dgtl Desgn Prncples 9
10 Boolen Algebr Two- nd Three-Vrble Propertes Couttve X Y = Y X X + Y = Y + X Assoctve X (Y Z) = (X Y) Z X + (Y + Z) = (X + Y) + Z Dstrbutve X (Y + Z) = X Y + X Z X + (Y Z) = (X + Y) (X + Z) Jnury, 22 ECE 52A - Dgtl Desgn Prncples
11 Boolen Algebr Absorpton (Splfcton) X + X Y = X X ( X + Y ) = X X Y X Y X X Y X+Y X Jnury, 22 ECE 52A - Dgtl Desgn Prncples
12 Boolen Algebr Cobnng (Splfcton) X Y + X Y = X (X + Y) (X + Y ) = X X Y X Y X Y X X Y X+Y X X+Y Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2
13 Boolen Algebr Redundnt Coverge (splfcton) X + X Y = X + Y X (X + Y) = X Y X Y X Y X X+Y X Y X +Y X Y X Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3
14 Boolen Algebr The Consensus Theore XY + X Z + YZ = XY + X Z X YZ XY X Z YZ Jnury, 22 ECE 52A - Dgtl Desgn Prncples 4
15 Boolen Algebr DeMorgn s Theore (X Y) = X + Y (X + Y) = X Y X Y X Y (X Y) X +Y (X+Y) X Y Jnury, 22 ECE 52A - Dgtl Desgn Prncples 5
16 Boolen Expressons Precedence of Opertons Order of evluton. NOT 2. AND 3. OR Or forced by prentheses Exple: F = b c + b + bc + b c =, b= nd c= NOT: AND: OR: Jnury, 22 ECE 52A - Dgtl Desgn Prncples 6
17 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Derve Logc Network, Krnugh Mp, Truth Tble nd Tng Dgr fro: F = b c + b + bc + b c 3 vrbles, lterls, 4 product ters Expresson s n Stndrd Su-of-Products for.e., the functon s the su (or logcl OR) or the four product (or logcl AND) ters The lterntve stndrd for s Product-of-Sus The expresson ples structure Drect relzton wth AND, OR nd NOT functons Jnury, 22 ECE 52A - Dgtl Desgn Prncples 7
18 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Logc Network F = b c + b + bc + b c Jnury, 22 ECE 52A - Dgtl Desgn Prncples 8
19 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Krnugh Mp F = b c + b + bc + b c bc bc b c b c b Jnury, 22 ECE 52A - Dgtl Desgn Prncples 9
20 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Note possble splfcton Redundnt coverge (elntes lterl) nd bsorpton (elntes product ter) bc b c b b Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2
21 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Truth Tble F = b c + b + bc + b c F c b
22 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Tng Dgr (Functonl Sulton) F = b c + b + bc + b c Input Output Jnury, 22 ECE 52A - Dgtl Desgn Prncples 22
23 Mnters nd Mxters Mnter A product ter whch contns ech of the n vrbles s fctors n ether copleented or uncopleented for s clled nter Exple for 3 vrbles: b c s nter; b s not Mxter A su ter whch contns ech of the n vrbles s fctors n ether copleented or uncopleented for s clled xter For 3 vrbles: +b+c s xter; +b s not Jnury, 22 ECE 52A - Dgtl Desgn Prncples 23
24 Mnters nd Mxters Mnter nd Mxter Expnson Three vrble exple: ( )' M nd ( M )' Jnury, 22 ECE 52A - Dgtl Desgn Prncples 24
25 Su-of-Products For Cnoncl Su-of-Products (or Dsjunctve Norl) For The su of ll nters derved fro those rows for whch the vlue of the functon s tkes on the vlue or ccordng to the vlue ssued by f. Therefore ths su s n fct n lgebrc representton of f. An expresson of ths type s clled cnoncl su of products, or dsjunctve norl expresson. Kohv Jnury, 22 ECE 52A - Dgtl Desgn Prncples 25
26 Mnters nd Mxters Truth Tble fro erler exple F = b c + b + bc + b c b c F M = M = 2 M 2 = 2 3 M 3 = 3 4 M 4 = 4 5 M 5 = 5 6 M 6 = 6 7 M 7 = 7 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 26
27 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 27 Su-of-Products Cnoncl Su-of-Products F = b c + b + bc + b c (,2,3,4,5) ' ' ' ' ' ' ' ' ' F c b c b bc bc c b F F F F
28 Product-of-Sus For Cnoncl Product-of-Sus (or Conjunctve Norl) For An expresson fored of the product of ll xters for whch the functon tkes on the vlue s clled cnoncl product of sus, or conjunctve norl expresson. Jnury, 22 ECE 52A - Dgtl Desgn Prncples 28
29 Product-of-Sus Cnoncl Product-of-Sus F = b c + b + bc + b c F ( F M )( ( M M )( )( M 2 M 2 )( M )( 2 3 M )( M F ( M 3 )( )( M )( M M )( )( M )( M M F ( b c')( ' b' c)( ' b' c') F M (,6,7) ) 5 5 )( 6 )( M M 6 6 )( 7 )( M 7 M ) 7 ) Jnury, 22 ECE 52A - Dgtl Desgn Prncples 29
30 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3 Generl Su-of-Product (SOP) nd Product-of-Sus (POS) Fors s the Boolen vlue of the functon n the th row of n n-vrble Truth Tble ) ' ( ' ' ) ( ) ( ) )( ( n n n n M F M M M M F F
31 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3 Equvlence of SOP nd POS Fors Mnter / Mxter Lsts (,2,3,4,5) (,6,7) ' ) ' ( ' ' (,6,7) (,2,3,4,5) ) ( M F M F nd M F M F exple exple n n n n
32 Functonlly Coplete Opertons A set of opertons s sd to be functonlly coplete (or unversl) f nd only f every swtchng functon cn be expressed entrely by ens of opertons fro ths set [Snce] every swtchng functon cn be expressed n cnoncl su-of-products [nd product-of-sus] for, where ech expresson conssts of fnte nuber of swtchng vrbles, constnts nd the opertons AND, OR nd NOT [ths set of opertons s functonlly coplete] Jnury, 22 ECE 52A - Dgtl Desgn Prncples 32
33 SOP Relzton wth NAND/NAND The NAND operton s functonlly coplete Jnury, 22 ECE 52A - Dgtl Desgn Prncples 33
34 POS Relzton wth NOR/NOR The NOR operton s functonlly coplete Jnury, 22 ECE 52A - Dgtl Desgn Prncples 34
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