# II. SOLUTIONS TO HOMEWORK PROBLEMS Unit 1 Problem Solutions

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1 II. SOLUTIONS TO HOMEWORK PROLEMS Unit Prolem Solutions (). () r r5= 6 (4). r = = r 6 r7 (2).72 6 ().52 6 (8).32. () r4 6 6 r6 (4).24 r 6 (3).84 6 (3).44 6 (7) = 64.E3 6 = E = = () r r2 (8). r = = () E.6 6 = E () = = /6 = () 2 E = = /8 = = /6 = = = /8 = = () = = r r5 () r2 6 r3 () () (2) () = = r r= 6 ().76 r5 6 (2) = () 5. 6 =. 2 = () 5. 6 = () E. 6 = E = =

2 .5 () () (Su) + (Multiply).5 (, ) See L p. 625 for solution..6,.7,.8 See L p. 625 for solution.. (). () r9 6 5 r (6) = = r3 6 r2 ().92 6 (4).72. () r5 = 6 6 (5).28 6 (4) = = () r2 6 6 r6 (4). 3.2 = 2.E 8 =. 2 2 E = 66.E 6 = E. (). 2 = = = /8 = (). 2 = = = /8 = = /6 = = = = /6 = () = /8 + 4/64 = r r (2) r r ().2 3 r 3 r ().63 3 () = () r r (2) r 4 4 r2 (3).84 r 4 (3) =

3 .2 ().4 ().4 () 52.4 = / + 4/2 = r r5 (8).46 9 r6 9 r (4) = = /64 = r r4 (7) r6 8 r5 (6) /64 = (or ) =. 2 (or. 2) 9 3/ r r5 (7) r5 8 r3 (6) = se 3 se () r5 8 8 r3 (5).6 r 8 (4) = =. 2.4 () r5 8 8 r5 (2).4 r 8 (3).2 9 3/32 = = = =. 2.5 () () (Sutrt) (Multiply).5() () (Su) (Mult) 7

4 .5() () (Su) (Mult).6 () () ().7 () Quotient ) Reminer.7() Quotient ) Reminer.7().8 ().8() Quotient ) Reminer Quotient ) Reminer Quotient ) Reminer.8().9 Quotient ) Reminer =

5 is possile, ut is not, euse there is no wy to represent 3 or 9. lternte Solutions: is not possile, euse there is no wy to represent 3 or is possile: () () () () = 83 lternte Solutions: () () () ().23 lternte.24 lternte Solutions: Solutions: = 94 () () () () () () 49 = lt.: = " " " r4 6 r3 (3).52 6 (8) = E.38 6 = E r7 6 r (2).96 6 (5) = 7. 6 = 7. 9

6 .26 () In 2 s omplement In s omplement ( ) + ( ) ( ) + ( ) () ( 2) ( 2).26 () In 2 s omplement In s omplement ( ) + ( 6) ( ) + ( 6) () ( 6) ( 6).26 () ( 8) + ( ) ( 8) + ( ) () ( 9) ( 9).26 () (2) (2).26 (e) ( ) + ( 4) ( ) + ( 4) () ( 5) ( 5).27 () In 2 s omplement In s omplement () In 2 s omplement In s omplement + + ().27 () + + () overflow overflow.27 () + + ().27 (e) In 2 s omplement In s omplement + + ().28 () In 2 s omplement In s omplement + + ().28 () () + + () overflow overflow.28 () + +

7 Unit 2 Prolem Solutions 2. See L p. 626 for solution. 2.2 () In oth ses, if =, the trnsmission is, n if =, the trnsmission is. 2.2 () In oth ses, if =, the trnsmission is YZ, n if =, the trnsmission is. Y Y Z Y Z 2.3 or the nswer to 2.3, refer to L p () = [( ) + ( )] + E + = + E () Y = (' + ( + )) + = (' + ) + = ( + ) + = + + = () ( + ) ( + ) (' + ) (' + E) = ( + ) (' + ) (' + E) y Th. 8 = (' + ) (' + E) y Th. 8 = ' + E y Th () (' + + ') (' + ' + ) (' + ') = (' + ' + ) (' + ') {y Th. 8 with = ' + '} = '' + '' + ' + '' + '' + ' = '' + '' + '' + '' 2.6 () + '' = ( + ') ( + ') 2.6 () W + WY' + ZY = (W + WY' + ZY) = ( + ') ( + ') ( + ') ( + ') = (W + ZY) {y Th. } = (W +Z) (W + Y) 2.6 () ' + E + E' = ' + E( +') = ' + E( +) = (' + E) (' + + ) = (' + E) ( + E) ( + E) (' + + ) ( + + ) ( + + ) 2.6 () YZ + W'Z + Q'Z = Z(Y + W' + Q') = Z[W' + (Y + Q')] = Z(W' + ) (W' + Y + Q') y Th (e) ' + '' + ' = ' ( + ') + ' = ' ( + ') + ' y Th. = (' + ') ( + ' + ') = (' + ') (' + ) ( + ' + ') y Th. = (' + ') ( + ') 2.6 (f) + + E = ( + + ) ( + + E) = ( + + ) ( + + ) ( + + E) ( + + E) 2.7 () ( ) ( E) ( ) = E pply seon istriutive lw (Th. 8) twie 2.7 () WYZ + VYZ + UYZ = YZ (W + V + U) y first istriutive lw (Th. 8) U V W E Y Z 2.8 () [()' + ']' = (')' = ( + ') 2.8 () [ + (' + )]' = '((' + ))' = + ' = '(' + (' + )') = '(' + ') = '' + '' 2.8 () (( + ') )' ( + ) ( + )' = (' + ') ( + )'' = (' + ')'' = ''

8 2.9 () = [( + )' + ( + ( + )')'] ( + ( + )')' 2.9 () G = {[(R + S + T)' PT(R + S)']' T}' = ( + ( + )')' y Th. with =(+(+)')' = (R + S + T)' PT(R + S)' + T' = '( + ) = ' = T' + (R'S'T') P(R'S')T = T' + PR'S'T'T = T' 2. () Y 2. () Y Y Y' Y' 2. () ' Y Z Y Z 2. () ' 2. (e) 2. (f) Y Z Y Z Y Y 2. () '' + ('')' = y Th () ( + ') + + ' = + ' y Th. 2. () + + '( + )' = + + ' y Th. 2. () (' + ')(' + E) = ' + 'E y Th (e) [' + ()' +E'] = ' + E' y Th (f) (' + )('E + )' + ('E + ) = 'E + + ' + y Th. 2.2 () ( + Y'Z)( + Y'Z)' = y Th () (W + ' + YZ)(W' + ' + YZ) = '+ YZ y Th () (V'W + )' ( + Y + Z + V'W) = (V'W + )' (Y + Z) y Th. 2.2 () (V' + W')(V' + W' + Y'Z) = V' + W' y Th. 2.2 (e) (W' + )YZ' + (W' + )'YZ' = YZ' y Th (f) (V' + U + W)(W + Y + UZ') + (W + UZ' + Y) = W + UZ' + Y y Th. 2.3 () = ' + + ( + ) = + + = 2.3 () 2 = '' + ' = ' + ' = ' + ' 2.3 () 3 = [( + )'][( + ) + ] = ( + )' ( + ) + ( + )' = ( + )' y Th. 5 & Th () Z = [( + )]' + ( + ) = [( + )]' + y Th. with Y = [( + ) ]' = '' + ' + ' 2.4 () ( + E + ) 2.4 () W + Y + Z + VU 2

9 2.5 () 2.5 () 2.5 () 2.5 () 2.5 (e) 2.5 (f) H'I' + JK = (H'I' + J)(H'I' + K) = (H' + J)(I' + J)(H' + K)(I' + K) + '' + ' = ( + '' + ') = [(' + )( + ') + '] = (' + + ')( + ' + ') ' + + E' = (' + + E') = [' + ( + E')] = (' + )(' + + E') ' + '' + E' = ' + '' + E' = ' ( + ') + E' = (' + E') ( + ' + E') = (' + E) (' + ') ( + E) ( + ' ) ( + ' + E) ( + ' + ') W'Y + W'' + W'Y' = '(WY + W') + W'Y' = '(W' + Y) + W'Y' = (' + W') (' + Y') (W' + Y + W') (W' + Y + Y') = (' + W') (' + Y') (W' + Y) ' + (' + E) = ' + ( + E)(' + E) = (' + + E)(' + ' + E) = ( + + E)(' + + E)( + ' + E)(' + ' + E) 2.6 () W + 'YZ = (W + ')(W + Y) (W + Z) 2.6 () VW + Y' + Z = (V++Z)(V+Y'+Z)(W++Z)(W+Y'+Z) 2.6 () '' + '' + 'E' = '(' + ' + E') 2.6 () = '[E' + (' + ')] = '(E' + )(E' + ' + ') + E' + ' = ( + E' + ') = [E' + ( + ')] = (E' + )(E' + + ') = ( + )( + E')( + ' + )( + ' + E') 2.7 () [(Y)' + (' + Y')'Z] = ' + Y' + (' + Y')'Z 2.7 () ( + (Y(Z + W)')')' = 'Y(Z + W)' = 'YZ'W' = ' + Y' + Z y Th. with Y = (' + Y') 2.7 () [(' + ')' + ('')' + '']' = (' + ')''( + ) = '' 2.7 () ( + ) ' + ( + )' = ' + ( + )' {y Th. with Y = ( + )'} = ' + '' 2.8 () = [(' + )']' + = [' + + '] + = () G = [()'( + )]' = ( + '') = 2.8 () H = [W''(Y' + Z')]' = W + + YZ 2.9 = (V + + W) (V + + Y) (V + Z) = (V + + WY)(V + Z) = V + Z ( + WY) y Th. 8 with = V W 2.2 () = + ' + ' + ' = + ' + ' (y Th. 9) = ( + ') + ' = (+ ) + ' (y Th. ) = + + ' = + ( + ') = + ( + ) = () Y + Z V eginning with the nswer to (): = ( + ) lternte solutions: = + ( + ) + 3 = + ( + )

10 2.2 () W Y Z W'Y WZ W'Y+WZ W'+Z W+Y (W'+Z)(W+Y) 2.2 () + +' (+)(+') ' +' 2.2 () Y Z +Y '+Z (+Y)('+Z) Z 'Y Z+'Y 2-2 () Y Z Y YZ 'Z Y+YZ+'Z Y+'Z 4

11 2.2 (e) Y Z +Y Y+Z '+Z (+Y)(Y+Z)('+Z) (+Y)('+Z) 2.22 ( + ) =, = [(+Y')Y] = Y' + Y, (Y) = + Y Unit 3 Prolem Solutions 3.6 () (W + ' + Z') (W' + Y') (W' + + Z') (W + ') (W + Y + Z) = (W + ') (W' + Y') (W' + + Z') (W + Y + Z) = (W + ') [W' + Y' ( + Z')] (W + Y + Z ) = [W + ' (Y + Z)] [W' + Y'( + Z')] = WY' ( + Z') + W'' (Y + Z) {Using ( + Y) (' + Z) = 'Y +Z with =W} = WY' + WY'Z' + W''Y + W''Z 3.6 () ( ) (' + ' + + ') (' + ) ( + ) ( + + ) = ( + + ) (' + ) ( + ) = ( + + ) (' + ) {Using ( + Y) (' + Z) = 'Y + Z with = } = ' + ' + ' = ' () + '' + '' + = + '(' + ') = (' + ) [ + (' + ')] {Using ( + Y) (' + Z) = 'Y + Z with =} = (' + ) [ + (' + ') (' + )] = (' + ) ( + ' + ') 3.7 () ''' + ' + ' + ' 3.8 = ' ('' + ) + (' + ') = ' [(' + ) ( + ')] + [(' + ') (' + )] {Using Y + 'Z = (' + Y) ( + Z) twie insie the rkets} = [ + (' + ) ( + ')] [' + (' + ') (' + )] {Using Y + 'Z = (' + Y) ( + Z) with = } = ( + ' + ) ( + + ') (' + ' + ') ( ' + ' + ) {Using the istriutive Lw} = [( ) + ] = ( + '' + ) = ('' + ) = (' + ) = ()' (' + ) + (' + )' = (' + ') (' + ) + (') = ' + ' + ' {Using ( + Y) ( + Z) = + YZ} = ' + ' + ' {Using + 'Y = + Y} 3.9 = ( Β) (Α ) is not vli istriutive lw. PROO: Let =, =, =. LHS: = = =. RHS: ( ) ( ) = ( ) ( ) = =. 5

12 3. () ( + W) (Y Z) + W' = ( + W) (YZ' + Y'Z) + W' = YZ' + Y'Z + WYZ' + WY'Z + W' 3. () ( ) + + = ' + ()' + + = ' + (' + ') + + = ' + ' + ' + + Using onsensus Theorem WYZ' + WY'Z + W' = ' + ' + ' ( onsensus term, eliminte ) = ' + ' + ' + (Remove onsensus term ) 3. () (' + ' + ') (' + + ') ( + + ) ( + + ) = (' + ' + ') ( + ' + ) (' + + ') ( + + ) ( + + ) onsensus term = (' + + ') ( + + ) = (' + ' + ') ( + ' + ) ( + + ) Removing onsensus terms 3. ( + ' + + E') ( + ' +' + E) (' + ' + ' + E') = [ + ' + ( + E') (' + E)] (' + ' + ' + E') = ( + ' + 'E' + E) (' + ' + ' + E') = ' + ( + 'E' + E) (' + ' + E') ' { onsensus term} = ' + ' + ' + E' + ''E' + 'E' + 'E' + 'E = ' + ' + ' + E' + ' +'E + 'E' = ' + ' + E' + ' + 'E' 3.2 ''E + ''' + E + = ''' + + 'E Proof: LHS: ''E + 'E + ''' + E + onsensus term to left-hn sie n use it to eliminte two onsensus terms = 'E + ''' + This yiels the right-hn sie. LHS = RHS 3.3 () (' + ' + ) (' + ) ( + ' + ') (' + + ) ( + ) = (' + + '') ( + ') = ( + '') + ('') {Using Y + 'Z = ( + Z)(' + Y) with = } = + '' + '' 3.3 () (' + ' + ') ( + + ') ( + ) (' + ) (' + + ) = [' + (' + ')] [ + ( + ')] = (' + ') + ' ( + ') = ' + ' + ' + '' 3.3 () (' + ' + ) ( + ') (' + + ') ( + ) ( + + ') = [' + (' + ) ( + ')] ( + ') = (' + + '') ( + ') {y Th. 4 with = } = ( + '') + '' {y Th. 4 with = } = + '' + '' 3.3 () 3.3 (e) ( + + ) (' + ' + ') (' + ' + ') ( + + ) = ( + + ) (' + ' + '') = ( ' + '') + '( + ) {y Th. 4 with = } = ' + '' + ' + ' ( + + ) ( + + ) (' + ' + ') (' + ' + ') = ( + + ) (' + ' + '') = (' + '') + '( + ) = ' + '' + ' + ' lt. soln's: ' + ' + '' + ' (or) ' + ' + ' + '' (or) ' + ' + '' + ' 3.4 () ' + '' + ' = '' + ' = ('' + ') = (' + '') {y Th. with Y = '} = ' + '' 6

13 3.4 () '' + ' + '' = '' + '( + ') = '' + '( + ) = '' + ' + ' 3.4 () 3.4 () 3.5 () 3.5 () 3.5 () ( + ') (' + ' + ) (' + + ') = ' + (' + ) ( + ') = ' + ( + ') = ' + (' + + ' + ) (' + ' + + E) (' +' + + E') = [' + ' + ( + ) ( + E) ( + E')] {y Th. 8 twie with = ' + '} = [' + ' + ( + )] = [' + ' + ] = '' + ' + + '' = ' ( + ') + ('' + ) = ' [( + ') ( + )] + [(' + ) ( + ')] {y Th. 4 twie with = n = } = [ + ( + ') ( + ) ] [' + (' + ) ( + ')] {y Th. 4 with = } = ( + + ') ( + + ) (' + ' + ) (' + + ') {y istriutive Lw} + '' + ''' + ' = ' (' + '') + ( + ') = ( + ' + '') (' + + ') {y Th. 4 with = } = ( + ' + ') ( + ' + ') (' + + ) (' + + ') + '' + '' + '' = ' [' + '] + [ + ''] = ' [( + ) (' + ')] + [( + ') ( + ')] = [ + ( + ) (' + ')][' + ( + ') ( + ')] = ( + + ) ( + ' + ') (' + + ') (' + + ') 3.5 () '' + '' + '' + = ('' + ) + ' (' + ') = ( + ') ( + ') + ' (' + ') ( + ) = [' + ( + ') ( + ')] [ + (' + ') ( + )] = (' + + ') (' + + ') ( + ' + ') ( + + ) 3.5 (e) WY + W'Y + WYZ + YZ' = WY ( + ' + Z) + YZ' = WY + YZ' = Y (W + Z') = Y (W + ) (W + Z') 3.6 () ( ) + '' = ()' + ' + '' = (' + ') + ' + '' = (' + ') + ' ( + ') = (' + ' + ') ( + ' + ) = (' + ' + ') ( + ' + ) ( + ' + ) 3.6 () ' ( ') + + ' = ' ['' + ] + + ' = ''' + ' + + ' = ''' + ( + ' + ') = ''' + ( + ' + ') = ''' + = + '' = (' + ) (' + ) 3.7 () 3.7 () ( Y) Z = (Y Z) Proof: LHS: Let Y =. Z = Z' + 'Z = ( Y) Z' + ( Y)' Z = ( Y ) Z' + ( Y) Z {y (3-8) on L p. 6) = ('Y + Y') Z' + (Y + 'Y') Z = 'YZ' + Y'Z' + YZ + 'Y'Z RHS: Let Y Z =. = ' + ' = (Y Z)' + ' (Y Z) = (Y Z) + ' (Y Z) = [YZ + Y'Z'] + ' [YZ' + Y'Z] = YZ + Y'Z' + 'YZ' + 'Y'Z LHS = RHS ( Y) Z = (Y Z) Proof: LHS: Let Y =. ( Z) = Z + 'Z' = ( Y) Z + ( Y)' Z' = ( Y ) Z + ( Y) Z' = (Y + 'Y') Z + (Y' + 'Y) Z' = YZ + 'Y'Z + Y'Z' + 'YZ' RHS: Let Y Z =. ( ) = + '' = (Y Z) + ' (Y Z)' = (Y Z) + ' (Y Z) = [YZ + Y'Z'] + ' [Y'Z + YZ'] = YZ + Y'Z' + 'Y'Z + 'YZ' LHS = RHS 3.8 () '' + ' + ' + ' + '' = '' + ' + ' + '' = ' + ' + '' 3.8 () W'Y' + WYZ + Y'Z + W'Y + WZ = W'Y' + WYZ + Y'Z + W'Y + WZ = W'Y' + WYZ + W'Y + WZ = W'Y' + W'Y + WZ 7

14 3.8 () ( + + ) ( + + ) (' + + ) (' + ' + ') = ( + + ) (' + + ) (' + ' + ') 3.8 () W'Y + WZ + WY'Z + W'Z' = W'Y + WZ + WY'Z +W'Z' + YZ = WY'Z + W'Z' + YZ YZ ( onsensus term) 3.8 (e) '' + '' + ' + ' + ' = '' + ' + ' 3.8 (f) ( + + ) ( + ' + ) ( + + ) (' + ' + ') = ( + + ) ( + ' + ) (' + ' + ') 3.9 Z = + E + + ' + 'E' = ( + + ' + 'E') + E = ( + E) (E ' + 'E') {y Th. 8 with = E} = ( + ) ( + E) ( + + ' + E + 'E') = ( + ) ( + E) (' + E + ' + + ) {Sine E + 'E' = E + '} = ( + ) ( + E) (' + E + ' + + ) {Sine ' + = ' + } = ( + ) ( + E) (' + E + ' +) {Sine + = } = ( + E) (' + E + ' + ) = ' + E + ' + + E + E' + E {eliminte onsensus term E; use + Y = where = E} 3.2 = ' + ' + + E = ' + + '' + E + = ( + ) (' + ) + ('' + E + ) = [( + ) (' + ) + ] [( + ) (' + ) + '' + E + ] = ( + ) (' + + ) ( + + '' + E + ) (' + + '' + E + ) + + ' = ( + ) (' + + ) ( + ) ( + + '' + E + ) (' + + ' + E + ) = ( + ) ( + ) (' + + ' + E + ) = ( + ) ( + ) (' + + ' + + E) = ( + ) ( + ) (' + + ' + ) = ( + ) (' + + ' + ) = (' + + ' ' + = ' + ' + + use onsensus, + Y = where = 3.2 'Y'Z' + YZ = ( + Y'Z') (' + YZ) = ( + Y') ( + Z') (' + Y) (' + Z) (Y + Z') = ( + Y') ( + Z') (' + Y) (' + Z) (Y + Z') = ( + Y') ( + Z') (' + Z) (Y + Z') = ( + Y') (' + Z) (Y + Z') lt.: (' + Y) (Y' + Z) ( + Z') y ing (Y' + Z) s onsensus in 3r step 3.22 () y + 'yz' + yz = y ( + 'z') + yz = y + yz' + yz = y + y = y lternte Solution: y + 'yz' + yz = y ( + 'z' + z) = y ( + z' + z) = y ( + ) = y 3.22 () 3.22 () y' + z + (' + y) z' = 'y + (' + y) {y Th. with Y = z} = y' + ' + y = + ' + y = + y = lt.: y' + z + (' + y) z' = (y' + z) + (y' + z)' = () (y' + z) ( + y') z = (y' + z + y'z) z = y'z + z + y'z = z + y'z lternte Solution: (y' + z) (+y') z = z ( + y') = z + zy' ' (' + ) + '' ( + ') +(' + ) ( + ') = '' + ' + '' + ''' + '' + = '' + '' + '' + ' Other Solutions: '' + + ''' + '' '' + + ''' + ' '' + + '' + '

15 3.22 (e) w'' + 'y' + yz + w'z' + 'z reunnt term = w'' + 'y' + yz + w'z' + 'z = 'y' + yz + w'z' + 'z Remove reunnt term = 'y' + yz + w'z' 3.22 (g) [(' + ' + ') ( + + ')]' + ''' + '' = ( + ') + '' (' + ) +''' + '' = + '+ ''' + '' + ''' + '' ' '' = + ''' + '' + ' = + '' + ' 3.22 (f) ' + ''+ 'E+ E'G+'E+''E = ' + 'E + E'G + 'E (onsensus) = ' + 'E + E'G 3.23 () ''' + ' + + '' + ' + '' 3.23 () = '' + ' + + ' onsensus = '' + ' WY' + (W'Y' ) + (Y WZ) 3.25 () = WY' + W'Y' + (W'Y')' ' + Y (WZ)' + Y'WZ = WY' + W'Y' + (W + Y) ' + Y (W' + Z') + Y'WZ = Y' + W' + 'Y + W'Y + YZ' + WY'Z + WY' = Y' + W' + 'Y + W'Y + YZ' + WY'Z + WY' = Y' + W' + W'Y + YZ' + WY' = + W' + W'Y + YZ' lternte Solutions: = W'Y + W' + WZ' + Y' = YZ' + W' + Y' + WY' = W' + 'Y + Z' + WY' = W' + Y' + WZ' + WY' ''' + + ' + '' + ' + '' = '' + + ' + ' = '' + + ' VLI: ' + ' + ' = ' ( + ') + ( + ') ' + ( + ') ' = ' + '' + ' + '' + ' + '' = ' + ' + ' lternte Solution: ' + ' + ' ll onsensus terms: ', ', ' We get = ' + ' + ' + ' + ' + ' = ' + ' + ' 3.25 () NOT VLI. ounteremple: =, =, =. LHS =, RHS =. This eqution is not lwys vli. In ft, the two sies of the eqution re omplements: [( + ) ( + ) ( + )]' = [( + ) ( + )]' = [ + + ]' = (' + ') (' + ') (' + ') 3.25 () VLI. Strting with the right sie, onsensus terms RHS = + '' + ' + ' + + ' = + '' + ' + ' + + ' = + '' + ' + ' + = LHS 3.25 () VLI: LHS = y' + 'z + yz' onsensus terms: y'z, z', 'y = y' + 'z + yz' + y'z + z' + 'y = y'z + z' + 'y = RHS 3.25 (e) NOT VLI. ounteremple: =, y =, z =, then LHS =, RHS =. This eqution is not lwys vli. In ft, the two sies of the equtions re omplements. LHS = ( + y) (y + z) ( + z) = [( + y)' + (y + z)' + ( + z)']' = ('y' + y'z' + 'z')' = [' (y' + z') + y'z']' =[(' + y'z') (y' + z' + y'z')]' = [(' + y') (' + z') (y' + z')]' (' + y') (y' + z') (' + z') 9

16 3.25 (f) VLI: LHS = ' + ' + '' () VLI: onsensus terms: ', = ' + ' + '' + + ' + ' + ' '' = RHS LHS = (' + Y') ( Z) + ( + Y) ( Z) = (' + Y') ('Z' + Z) + ( + Y) ('Z + Z') = 'Z' + 'YZ' + Y'Z + 'YZ + Z' + YZ' = 'Z' + (Y' + 'Y)Z + Z' = Z' + Z( Y) = Z' + ( Y) = RHS 3.26 () LHS = (W' + + Y') (W + ' + Y) (W + Y' + Z) = (W' + + Y') (W + (' + Y) (Y' + Z)) = (W' + + Y') (W + ('Y' + YZ)) = (W' ('Y' + YZ) + W ( + Y ')) = W''Y' + W'YZ + W + WY' onsensus terms: 'Y' YZ = W''Y' + W'YZ + W + WY' + YZ + 'Y' = W''Y' + W''Z + W'YZ + YZ + W + WY' + 'Y' = W''Z + W'YZ + YZ + W + 'Y' = W'YZ + YZ + W + 'Y' 3.26 () LHS = + ''' + '' + = ( + ) + '' ( + ') = ( + ' ( + ')) (' + ( + )) = ( + ') ( + + ') (' + ) (' + + ) = ( + ') ( + + ') (' + ) (' + + ) ( + ' + ) onsensus: + ' + = ( + ') ( + + ') (' + ) ( + ' + ) = ( + ') (' + ) ( + ' + ) = RHS 3.27 () VLI. [ + = ] [' ( + ) = '()] [ + = ] [' + ' = '] 3.27 () NOT VLI. ounteremple: =, = = n = then LHS = + = RHS = = = LHS ut + = + = ; = + The sttement is flse () VLI. [ + = ] [( + ) + = () + ] [ + = ] [ + + = + ] 3.27 () NOT VLI. ounteremple: =, = = n = then LHS = + + = RHS = + = = LHS ut + = + = The sttement is flse () '' + + ' + ' + ''' + ' onsensus terms: () '' using '' + ' (2) ' using '' + (3) using ' + (4) '' using ''' + ' Using, 2, 3: '' + + ' + ' + ''' + ' + '' + ' + = '' + + ' (Using the onsensus theorem to remove the e terms sine the terms tht generte them re still present.) 3.28 () ''' + ' + '' + ' onsensus terms: () '' using ''' + ' (2) ' using '' + ' (3) ''' using ''' + '' (4) '' using ''' + ' (5) ' using ' + ' Using : ''' + ' + '' + ' + ', whih is the minimum solution. 2

17 Unit 4 Prolem Solutions See L p. 628 for solution. E y z 4.2 () Y = ''''E' + '''E' + ''E' (less thn gpm) + (t lest gpm) () Z = ''E' + 'E' + E' (t lest 2 gpm) + + (t lest 3 gpm) + (t lest 4 gpm) + (t lest 5 gpm) 4.3 = m(, 4, 5, 6); 2 = m(, 3, 4, 6, 7); + 2 = m(, 3, 4, 5, 6, 7) Generl rule: + 2 is the sum of ll minterms tht re present in either or 2. 2 n Proof: Let = i m i ; 2 = j m j ; + 2 = i m i + j m j = m + m + 2 m Σi = 2 n Σj = 2 n Σi = 2 n Σj = 2 n + m + m + 2 m = ( + ) m + ( + ) m + ( ) m = ( i + i ) m i Σi = 4.4 () 4.4 () 2 2n = 2 22 = 2 4 = 6 y z z z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 z z z 2 z 3 z 4 z 5 'y' 'y ' y' y' 'y+y' '+y' y 'y'+y y '+y +y' +y 4.5 E Z lternte Solutions E Z 3 4 These truth tle entries were me on't res euse = n = n never our 2 These truth tle entries were me on't res euse when is, the output Z of the OR gte will e regrless of its other input. So hnging n E nnot ffet Z. 3 These truth tle entries were me on't res euse when n E re oth, the output Z of the OR gte will e regrless of the vlue of. 4 These truth tle entries were me on't res euse when one input of the N gte is, the output will e regrless of the vlue of its other input. 4.6 () Of the four possile omintions of & 5, = n 5 = gives the est solution: = ''' + '' + ' + = '' () y inspetion, G = when oth on t res re set to. 2

18 4.7 () Etly one vrile not omplemente: = '' + '' + '' = m(, 2, 4) 4.8 () (,,, ) = m(,, 2, 3, 4, 5, 6, 8, 9, 2) Refer to L for full term epnsion 4.7 () Remining terms re mterms: = M(, 3, 5, 6, 7) = ( + + ) ( + ' + ') (' + + ') (' + ' + ) (' + ' + ') 4.8 () (,,, ) = Π M(7,,, 3, 4, 5) Refer to L for full term epnsion 4.8 = 2 = 2 2 = 2 3 = 2 = 2 = 2 2 = = 3 > 2 2 = 2 2 = = 4 > = 6 > 2 3 = 2 3 = 3 > = 6 > = 9 > () 4.9 () 4.9 () 4.9 () = ' + ' ( + ') ( + ') = ' + ' + '' + '' + '''; = m(,, 4, 5, 6) Remining terms re mterms: = M(2, 3, 7) Mterms of re minterms of ': ' = m(2, 3, 7) Minterms of re mterms of ': ' = M(,, 4, 5, 6) 4. (,,, ) = ( + + ) (' + ) (' + ' + ') ( + + ' + ') = ( ) ( + + ' + ) (' + + ' + ') (' + ' + ' + ) (' + ' + ' + ') ( + + ' + ') = ( ) ( + + ' + ) (' ) (' ') (' + ' + + ) (' + ' + + ') (' + ' + ' + ) (' + ' + ' + ') ( + + ' + ') 4. () = m(, 4, 5, 6, 7,, ) 4. () = M(, 2, 3, 8, 9, 2, 3, 4, 5) 4. () ' = m(, 2, 3, 8, 9, 2, 3, 4, 5) 4. () ' = M(, 4, 5, 6, 7,, ) 4. () ifferene, i = i y i i ; i+ = i i ' + i 'y i + i y i 4. () i = s i ; i+ is the sme s i+ with i reple y i ' i y i i i+ i 4.2 See L p. 629 for solution. 22

19 4.3 Z Z = '''' + ''' + ''' + ' + + ' = ''' + + ''' + ' = ''' + + ''' + ' + + ''' (e onsensus terms) Z = ''' ''' ' ' ' ' ' ' Z 4.4 Z Z = '' + '' + ' + ''' + '' + '' + ' = ' + '' + ' + '' = ' + ' + '' + ' (e onsensus terms) Z = ' + ' + ' ' ' ' Z 4.5 () The uzzer will soun if the key is in the ignition swith n the r oor is open, or the set elts re not fstene. K S' The two possile interprettions re: = K + S' n = K( + S') 4.5 () You will gin weight if you et too muh, or you o not eerise enough n your metolism rte is too low. W E' M The two possile interprettions re: W = ( + E') M n W = + E'M 4.5 () The speker will e mge if the volume is set too high n lou musi is plye or the stereo is too powerful. V M S The two possile interprettions re: = VM + S n = V (M + S) 4.5 () The ros will e very slippery if it snows or it rins n there is oil on the ro. V S R O The two possile interprettions re: V = (S + R) O n V = S + RO 4.6 Z = Z = (E + ''''E')'; Y = '''E 4.8 () 3 = 6 = ; = '''E'G 4.8 () = ; Y = '''E'G' 4.8 () = 2 ; 64 = 2 ; 3 = 2 ; 27 = 2 ; 32 = 2 ; Z = ('')' = + 23

20 4.9 2 = M(, 3, 4, 5, 6, 7). Generl rule: 2 is the prout of ll mterms tht re present in either or 2. Proof: 2 n Let = ( i + M i ); 2 = ( j + M j ); 2 = ( i + M i ) ( j + M j ) Πi = 2 n Πj = 2 n Πi = 2 n Πj = = ( + M ) ( + M ) ( + M ) ( + M ) ( 2 + M 2 ) ( 2 + M 2 )... = ( + M ) ( + M ) ( M 2 )... 3 n = ( i i + M i ) Πi = Mterm M i is present in 2 iff i i =. Mterm M i is present in iff i =. Mterm M i is present in 2 iff j =. Therefore, mterm M i is present in 2 iff it is present in or G H J () (,,, ) = m(5, 6, 7,,, 3, 4, 5) = M(,, 2, 3, 4, 8, 9, 2) () G (,,, ) = m(, 2, 4, 6) = M(, 3, 5, 7, 8, 9,,, 2, 3, 4, 5) () H (,,, ) = m(7,, 3, 4, 5) = M(,, 2, 3, 4, 5, 6, 8, 9,, 2) () J (,,, ) = m(4, 8, 2, 3, 4) = M(,, 2, 3, 5, 6, 7, 9,,, 5) 4.22 f () f = m(, 2, 4, 5, 6,,, 2, 4, 5) () f = M(, 3, 7, 8, 9, 3) () f ' = m(, 3, 7, 8, 9, 3) () f '= M(, 2, 4, 5, 6,,, 2, 4, 5) You n lso work this prolem lgerilly, s in prolem You n lso work this prolem using truth tle, s in prolem f(,, ) = ' ( + ') = ' + '' = ' ( + ') + ' ( + ') ' = ' + '' + '' + ''' m 3 m 2 m 2 m 4.2 () f = m(, 2, 3) 4.2 () f = M(, 4, 5, 6, 7) 4.2 () f ' = m(, 4, 5, 6, 7) 4.2 () f ' = M(, 2, 3) 4.23 () (,,, ) = m(3, 4, 5, 8, 9,,, 2, 4) 4.23 () = '' + ''' + '' + ''' + '' + '' + ' + '' + ' (,,, ) = M(,, 2, 6, 7, 3, 5) = ( ) ( ') ( + + ' + ) ( + ' + ' + ) ( + ' + ' + ') (' + ' + + ') (' + ' + ' + ') 4.24 () (,,, ) = m(, 3, 4, 7, 8, 9,, 2, 3, 4) = '''' + '' + ''' + ' + ''' + '' m m 3 m 4 m 7 m 8 m 9 + ' + '' + ' + ' m m 2 m 3 m () (,,, ) = M(, 2, 5, 6,, 5) = ( ') ( + + ' + ) ( + ' + + ') ( + ' + ' + ) M M 2 M 5 M 6 (' + + ' + ) (' + ' + ' + ') M M 5 24

21 4.25 () If on't res re hnge to (, ), respetively, = ''' + + '' + ' = '' +, 4.25 () If on't res re hnge to (, ), respetively 3 = ( + + ) ( + + ') = () If on't res re hnge to (, ), respetively 2 = '''+ '' + '' + ' = ' 4.25 () If on't res re hnge to (, ), respetively 4 = ''' + ' + '' + = '' E Z These truth tle entries were me on't res 2 euse = n 2 = n never 2 our. 2 These truth tle entries were me on't res 2 euse when one input 2 of the OR gte is, the output will e regrless of the vlue of its other 2 input () G (,, ) = m(, 7) = M(, 2, 3, 4, 5, 6) 4.27 () G 2 (,, ) = m(,, 6, 7) = M(2, 3, 4, 5) 4.28 's Y Z () = Y = '' + '' + '' + ' + '' + '' + ' + '' + ' + ' Z = ''' + ''' + ''' + ' + ''' + ' + ' + ' 4.29 W Y Z () = ''' + ''' + '' + ''' + '' + '' + ' + ''' + '' + '' + ' + '' + ' + ' + Y = '''' + ' + ' + ' + Z = '''' + '' + '' + '' + '' + '' + ' + '' () Y = ( ) ( ') 4.29 () Y = ( ') ( + + ' + ) ( + + ' + ) ( + ' + + ) ( + + ' + ') ( + ' + + ) (' ) (' + ' + ' + ') ( + ' + + ') ( + ' + ' + ) (' ) (' ') (' + + ' + ) (' + + ' + ') (' + ' + + ) Z = ( ) ( + ' + + ') ( + ' + ' + ) (' ') (' + + ' + ) (' + ' + + ) (' + ' + ' + ') Z = ( ') ( + + ' + ) ( + ' + + ) ( + ' + ' + ) (' ) (' + ' + + ') (' + ' + ' + ) 25

22 4.3 S T U V W Y Z 4.3 S T U V W Y Z 5 = 4 + = 5 = = = = = = = = = = = = = = = = = =37 Note: Rows through hve on't re outputs. Note: Rows through hve on't re outputs. S =, T =, U =, V =, W =, =, Y =, Z = S =, T =, U = + +, V = ' + '' +, W = '' +, = '' + ', Y = ' + '' +, Z = 4.32 Notie tht the sign it 3 of the 4-it numer is etene to the leftmost full er s well. S 4 S 3 S 2 S S Y 4 Y 3 Y Y Y 4.33 Y Sum out Y Sum out S 3 S 2 S S H.. 2 H.. H.. H

23 Unit 5 Prolem Solutions 5.3 () f 5.3 () 5.3 () f2 f3 r e f s t 5.3 () f4 y z f = '' + ' + ' f2 = 'e' + 'f + e'f f3 = r' + t' f4 = 'z + y + z' 5.4 () 5.4 () 5.4 () = ' + ' + + ' + '' = ' + ' + = ( + '+ ') ( + + ') 5.5 () 5.5 () See L p. 63 for solution. 2 2 lt: Z = ' ' 2 + ' 2 ' + 2 ' 2 ' ' 2 ' 2 Z = ' ' 2 + ' 2 ' + 2 ' 2 ' ' 2 ' Z = ' ' 2 + ' 2 ' + 2 ' 2 ' ' () * * * * 5.6 () * * * lt: = ''' + ' + ' + ' + ' = ''' + ' + ' + ' + ' (*) inites minterm tht mkes the orresponing prime implint essentil. lt: = ' + '' + + '' = ' + '' + + ' (*) Inites minterm tht mkes the orresponing prime implint essentil. ' m 5 ; ''' m ; ' m ;' m 2 m 3 or m 5 ; ' m 3 ; '' m 8 or m 27

24 5.6 () * 5.7 () * * * = '' + ' + '' (*) Inites minterm tht mkes the orresponing prime implint essentil. lt: = ''' + ' + ''' + ' + ''' = ''' + ' + ''' + ' + '' '' m 2 ; '' m 6 ; ' m or m 5.7 () 5.7 () 5.7 () = '' + ''' + = ''' + ' + ' + ' + ' = () = ('+ ') ('+ ') ( + + ) ('+ + ) = ' ' + ' + ' ' 5.8 () lt: = ('+ ) ('+ ') ( + ) ('+ ) = ('+ ) ('+ ') ( + ) ('+ ') = '' + '' + 28

25 5.9 () E = (' + ' + + E) (' + + ' + ') ( + ' + ' + E) (' + + E) ( + ' + ) (' E') (' + ' + ' + E') E lt: = ''E + '' + ' E + ' ' + ' E + E' + ''E' + '' = ''E + '' + ' E + ' ' + ' E + E' + ''E' + ''E' 5.9 () E E = (' + ' + E) (' + ' + + E) ( + ' + E') ( + + ' + E) ( + + ) (' + + E') lt: = ' ' + ' E' + E + ''' + ' E' + ' E = ' ' + ' E' + E + ''E' + ' + ''E = ' ' + ' E' + E + ''' + ' E' + ''E = ' ' + ' E' + E + ''E' + ' E' + ''E 29

26 5. () e 5. () e Essentil prime implints: ''E' (m 6, m 24 ), 'E' (m 4 ), E (m 3 ), ''E (m 3 ) Prime implints: ''E, ''E', 'E, 'E', E, '', 'E, ''E', '' 5. E = ('+ + ') ('+ '+ E ) ( + '+ E') ( + + E') ( ) ('+'+ + ) ( + + E') 5.2 () = '' + ' + ' + = M(,, 9, 2, 3, 4) = ( ) ( ') (' + ' + + ) (' + ' + + ') (' + ' + ' + ) (' ') lt: = ('+ + ') ('+ '+ E ) ( + '+ E') ( + + E') ( ) ('+'+ + E ) ( + + E') ' 5.2 () () ' = ''' + ' + ' = ' + ' + ' + '' Minterms m, m, m 2, m 3, m 4, m, n m n e me on t res, iniviully, without hnging the given epression. However, if m 3 or m 4 is me on t re, the term ' or the term ' (respetively) is not neee in the epression. = ('+ '+ ) ( + + ) ('+ + ') 3

27 5.4 () 5.4 () () R E S T 5.4 () 5.4 (e) N P Q = ' + ' 2 = E' + E' + = T' + R = + ' = N'P + N Q 5.4 (f) 5.5 () f Y Z 5.5 () G E 5.5 () p q r 5.5 () s t u = Y' + 'Z' + Z f = '' + + ' f = '' + + G = E ' + 'E' G = E ' + ' G = E ' + E' = p'r + q'r' + p q = p'q' + p r' + q r = s' 5.5 (e) 5.5 (f) e f 5.6 () 5.6 () = '' + ' + = '' + ' + g = 'e' + f ' 5.6 () = '' + ' + = ('+ ) ( + '+ ') ('+ ) ('+ + ') 5.7 (), () 5.7 () = ' + ' + ' ' lt: = ('+ + ) ('+ '+ ') ('+ '+ ) = ('+ + ) ('+ '+ ') ('+ '+ ') 3

28 5.8 () 5.8 () 2 2 Z 2 2 = ( + 2+ ) ( + + 2) ( + 2'+ '+ 2') ( '+ 2'+ + 2') ( '+ '+ 2 ) ( ) 5.9 () 5.9 () = ' + ' + ' + ''' + ' = ''' + ''' + ''' lt: = ' + ' + ' + ''' + ' lt: = ''' + ''' + '' 5.9 () 5.9 () W Y Z 5.9 (e) = '' + ' + ' lt: = 'Y' + W'Z + Y'Z + W Z' = 'Y' + W Y' + W'Z + W Z' = 'Y' + W Y' + W'Z + W ' = '' + ' + ' () 5.2 () 5.2 () = ' + ' ' + ' + = '' + ' + ' + + = '' + ' + ' 32

29 5.2 () 5.2 () 5.2 () = ' ' + '' + = ('+ ) ('+ + ') ( + '+ ) ( + ) = ('+ ) ('+ + ') ( + ) ('+ ) 5.22 () lt: = ('+ ) ('+ + ') ( + ) ('+ ) 5.22 () = ('+ '+ ) ('+ ') ( + + ) w y z = ' + '' + ' + ' w y z = 'y' + w'z + y'z + w z' = (w + '+ z ) (w + y'+ z ) (w'+ y'+ z') lt: = 'y' + w y' + w'z + w z' = 'y' + w y' + w'z + w ' lt: = (w + '+ z ) (w + y'+ z ) (w'+ '+ y') () = '' + ' + ' Notie tht = n never our, so minterms 5 n 5 re on t res. = '' + ' + ' + = M(,, 9, 2, 3, 4) = ( ) ( ') (' ' ) (' + ' + + ) (' + ' + + ') (' + ' + ' + ) 33

30 5.24 () 5.24 () ' = ' + ''' + ' = ('+ '+ ) ( + + ) ('+ + ') Prime implints for f ': 'e, '', 'e', 'e, ''e', ''e, ''e Prime implints for f: ''e', e, 'e', e',, e', ''e, ''e, '', 'e 5.27 or : ''e', 'e, 'e', '', 'e, ''e, ''e 5.28 () = + E * * * (*) Inites minterm tht mkes the orresponing prime implint essentil. ''' m ; 'e' m 28 ; ''e m 25 ; '' m 2 * or G: 'e, ', 'e', e, 'e, ''', ''e' 5.28 () E ''', 'e', ''e, '', 'e', 'e', 'e', '''e, ''e, '', 'e', 'e', 'e' 5.29 () E = '' + ' ' + '' + ''E' + ' + E + ''E lt: = '' + ' ' + '' + ''E' + ' + E + ''E 34

31 5.29 () E lt: = '''E + ' '' + E + ''E' + E + 'E' + '''E + ' E = '''E + ' '' + E + ''E' + E + 'E' + ' 'E + ' E = '''E + ' '' + E + ''E' + E + 'E' + ' 'E + E 5.3 E = '''' + 'E + ' + 'E' + ' + ''E + 'E 5.3 E = ''E' + '' + E' + '''E' + ''E + ''E 5.32 () w y z v = v' y'z' + 'y'z + v z + w 'y z' + v w 5.32 () w 5.33 () y z E v = ( + y + z) (v + y' + z') (v + ' + z') (v + ' + y') (v' + w + z) lt: = ( + + E ) ('+ ) ( + ') ( + '+ '+ E') = ( + + E ) ('+ ) ( + ') ( + '+ '+ E') 35

32 5.33 () E lt: = ( + ') ( + '+ E') ('+ + E') ('+ '+ '+ ) ( + + E ) ( + '+ E ) = ( + ') ( + '+ E') ( + '+ ) ('+ + E') ('+ '+ '+ ) ( + '+ E ) 5.34 () w y z v lt: = (v'+ w'+ '+ y + z') (w + y'+ z') (v + y') (w + + y ) (v'+ + y + z ) (w'+ + y') = (v'+ w'+ '+ y + z') (w + y'+ z') (v + y') (w + + y ) (v'+ w'+ + z ) (w'+ + y') = (v'+ w'+ '+ y + z') (w + y'+ z') (v + y') (w + + y ) (v'+ w'+ + z ) ( + y'+ z') 5.34 () 5.35 () E = ( + + E) (' + ' + ') (' ) ( + ' + ) ( + ' + E) ( + + E) = ' + + ' hnging m to on't re removes ' from the solution () 5.36 () w y z v m 4 = v' y' + v'w z' + y z + v w''y' + v w'y z' + w' z m 8 = ' + ' + ' + '' m 4, m 3, or m4 hnge the minimum sum of prouts, removing '', ', or ', respetively. m 3 = v' y' + v'w z' + y z + v w''z' + v w' y + w'y'z = v' y' + v'w z' + y z + v w''y' + v w'y z' + w'y'z = v' y' + v'w z' + y z + v w''z' + v w'y z' + w'y'z 5.36 () 36 v'wz' m 8 ; yz m 3 ; v'y' m 4

33 Unit 6 Prolem Solutions 6.2 () ü, 5 - '' 5 ü, 9 - '' 9 ü 5, 7 - ' 2 ü 9, - ' 7 ü 2, 4 - ' ü 7, 5-4 ü, 5-5 ü 4, 5 - Prime implints: '', '', ', ', ',,, 6.2 () ü, - ''', 3, 5, 7 -- ' ü, 8 - ''', 5, 3, ü, 3 -ü 6, 7, 4, ü, 5 -ü 6, 4, 7, ü 8, - '' 6 ü 3, 7 -ü ü 5, 7 -ü 7 ü 6, 7 -ü 4 ü 6, 4 -ü 5 ü, 4 - ' 7, 5 -ü 4, 5 -ü Prime implints: ''', ''', '', ', ', 6.3 () , 5 '', 9 '' 5, 7 ' 9, ' 2, 4 ' 7, 5, 5 4, 5 f = ' + '' + ' + f = ' + '' + ' () , 3, 5, 7 ' 6, 7, 4, 5, ''', 8 ''' 8, '', 4 ' f = ' + + ''' + '' f = ' + + ''' + '' f = ' + + ''' + ' 37

34 6.4 ü, 3 -ü, 3, 5, 7 -- ' 2 ü, 5 -ü, 5, 3, ü, 9 -ü, 5, 9, 3 -- ' 3 ü 2, 3 -ü, 9, 5, ü 2, 6 -ü 2, 3, 6, 7 -- ' 6 ü 2, - '' 2, 6, 3, ü 4, 5 -ü 4, 5, 6, 7 -- ' ü 4, 6 -ü 4, 5, 2, 3 -- ' 2 ü 4, 2 -ü 4, 6, 5, ü 3, 7 -ü 4, 2, 5, ü 5, 7 -ü 5, 7, 3, ü 5, 3 -ü 5, 3, 7, , 7 -ü 9, 3 -ü 2, 3 -ü 7, 5 -ü 3, 5 -ü 3, 5 -ü Prime implints: '', ', ', ', ', ', , 3, 5, 7 ', 5, 9, 3 ' 2, 3, 6, 7 ' 4, 5, 6, 7 ' 4, 5, 2, 3 ' 5, 7, 3, 5 2, '' f = ' + '' + ' + ' f = ' + '' + ' + ' f = ' + '' + ' + ' 6.5 ü, 5 -ü, 5, 9, 3 -- ' 4 ü, 9 -ü, 9, 5, ü 4, 5 -ü 4, 5, 2, 3 -- ' 5 ü 4, 2 -ü 4, 2, 5, ü 8, 9 -ü 5, 7, 3, ü 8, 2 -ü 5, 3, 7, ü 5, 7 -ü 8, 9, 2, 3 -- ' ü 5, 3 -ü 8, 2, 9, ü 9, -ü 9,, 3, ü 9, 3 -ü 9, 3,, ü 2, 3 -ü 2, 3, 4, , 4 -ü 2, 4, 3, , 5 -ü, 5 -ü 3, 5 -ü 4, 5 -ü Prime implints: ', ',, ',, 38

35 6.5 (ont) P (, 5, 9, 3) ' P2 (4, 5, 2, 3) ' P3 (5, 7, 3, 5) P4 (8, 9, 2, 3) ' P5 (9,, 3, 5) P6 (2, 3, 4, 5) (P + P4 + P5) (P2 + P4 + P6) (P + P2 + P3 + P4 + P5 + P6) (P3 + P5 + P6) = (P4 + PP2 + PP6 + P2P5 + P5P6) (P3 + P5 + P6) = P3P4 + P4P5 + P4P6 + PP2P3 + PP2P5 + PP2P6 + PP3P6 + PP5P6 + PP6 + P2P3P5 + P2P5 + P2P5P6 + P3P5P6 + P5P6 = = (' + ) or ( + ') or ( + ') or ( + ) or ( + ') or ( + ') P4 P3 P5 P2 P5 P4 P6 P5 P6 P4 P6 P 6.6 () E = E = E E = MS + EMS = ' + ''' + ' + E ('' + ) or E ('' + ) MS = ''' + ' + ' 6.6 () E = = G = E = ; = G = E MS = '' + MS = '' + G E MS = '' = ; E = G = G = ; E = = MS = '' + ' MS = '' + Z = '' + + E ('' + ') + () + G (') MS = 2 MS 3 = ' or ' or 39

36 6.7 () ü, 4 - ''' 4 ü 4, 5 - '' 3 ü 3, 7 - ' 5 ü 3, - ' 9 ü 5, 7 - ' 7 ü 5, 3 - ' ü 9, - ' 3 ü 9, 3 - ' Prime implints: ''', '', ', ', ', ', ', ' 6.7 () 2 ü 2, 6 - '' 4, 5, 2, 3 -- ' 4 ü 2, - '' 4, 2, 5, ü 4, 5 -ü 9,, 3, ü 4, 6 - '' 9, 3,, ü 4, 2 -ü ü 5, 3 -ü 2 ü 9, -ü ü 9, 3 -ü 3 ü, - ' 5 ü 2, 3 -ü, 5 -ü 3, 5 -ü Prime implints:, ', '', '', '', ' 6.8 () , 4 ''' 4, 5 '' 3, 7 ' 3, ' 5, 7 ' 5, 3 ' 9, ' 9, 3 ' f = ''' + ' + ' + ' f = ''' + ' + ' + ' 6.8 () , 6 '' 2, '' 4, 6 '', ' 4, 5, 2, 3 ' 9,, 3, 5 f = ' + + '' + '' f = ' + + '' + ' f = ' + + '' + '' 6.9 () ü, 3 -ü, 3, 9, -- ' 2 ü, 9 -ü, 9, 3, -- 4 ü 2, 3 -ü 2, 3,, -- ' 3 ü 2, -ü 2,, 3, -- 9 ü 4, 2 - '' 3, 7,, 5 -- ü 3, 7 -ü 3,, 7, ü 3, -ü 9,, 3, ü 9, -ü 9, 3,, 5 -- ü 9, 3 -ü,, 4, ü, -ü, 4,, ü, 4 -ü 2, 3, 4, ü 2, 3 -ü 2, 4, 3, , 4 -ü 7, 5 -ü, 5 -ü 3, 5 -ü 4, 5 -ü 4 Prime implints: '', ', ',,,,

37 6.9 () (ont) , 2 '', 3, 9, ' 2, 3,, ' 3, 7,, 5 9,, 3, 5,, 4, 5 2, 3, 4, 5 f = ' + '' + + ' + f = ' + '' f = ' + '' () ü, -ü,, 8, 9 -- '' ü, 8 -ü, 8,, ü, 5 -ü, 5, 9, 3 -- ' 5 ü, 9 -ü, 9, 5, ü 8, 9 -ü 8, 9,, -- ' 9 ü 8, -ü 8,, 9, -- ü 8, 2 -ü 8, 9, 2, 3 -- ' 2 ü 5, 7 - ' 8, 2, 9, ü 5, 3 -ü ü 6, 7 - ' 3 ü 9, -ü 9, 3 -ü, -ü 2, 3 -ü Prime implints: ', ', '', ', ', ' , 7 ' 6, 7 ',, 8, 9 '', 5, 9, 3 ' 8, 9,, ' 8, 9, 2, 3 ' f = ' + '' + ' + ' 6.9 () f = ' + + '' + + f = ' + + '' + + f = ' + + '' + + ' 6. Prime implints: ', ', ', ', ', ''' f = ' + ' + ' + ''' + ' f = ' + ' + ' + ''' + ' 4

38 6. ü, 2 -ü, 2, 4, 6 --ü, 2, 4, 6, 8,, 2, 'E' 2 ü, 4 -ü, 2, 8, --ü, 2, 8,, 4, 6, 2, ü, 8 -ü, 2, 6, 8 -- ''E', 4, 8, 2, 2, 6,, ü, 6 -ü, 4, 2, ü 2, 6 -ü, 4, 8, 2 --ü 6 ü 2, -ü, 8, 2, -- 9 ü 2, 8 -ü, 8, 4, 2 -- ü 4, 6 -ü, 6, 2, ü 4, 2 -ü 2, 6,, 4 --ü 8 ü 8, 9 -ü 2,, 6, ü 8, -ü 4, 6, 2, 4 --ü ü 8, 2 -ü 4, 2, 6, ü 6, 8 -ü 8, 9,, -- '' 4 ü 6, 7 - '' 8, 9, 2, 3 -- '' 9 ü 6, 4 -ü 8,, 9, -- 2 ü 9, -ü 8,, 2, 4 --ü 29 ü 9, 3 -ü 8, 2, 9, ü, -ü 8, 2,, 4 --, 4 -ü 2, 3 -ü 2, 4 -ü 8, 9 - '' 3, 29 - 'E 4, 3 - E' 2, 29 - 'E , 7 '' 8, 9 '' 3, 29 'E 4, 3 E' 2, 29 'E, 2, 6, 8 ''E' 8, 9,, '' 8, 9, 2, 3 '', 2, 4, 6, 8,, 2, 4 'E' = E' + '' + ''E' + '' + '' + 'E + 'E' 42

39 6.2 () ü, -ü,, 2, 3 --ü,, 2, 3, 8, 9,, --- '' ü, 2 -ü,, 8, 9 --ü,, 8, 9, 2, 3,, ü, 8 -ü, 2,, 3 --, 2, 8,,, 3, 9, ü, 3 -ü, 2, 8, --ü 3 ü, 9 -ü, 8,, ü, 7 -ü, 8, 2, -- 9 ü 2, 3 -ü, 3, 9, --ü ü 2, 6 - ''E', 9, 3, -- 7 ü 2, -ü, 9, 7, ''E 2 ü 8, 9 -ü, 7, 9, ü 8, -ü 2, 3,, --ü 2 ü 3, -ü 2,, 3, ü 9, -ü 8, 9,, --ü 28 ü 9, 25 -ü 8,, 9, ü, -ü 3 ü 7, 2 - ''E 3 ü 7, 25 -ü 2, 2 - '' 2, 28 - 'E' 2, 23 - 'E 28, 3 - E' 23, 3 - E 3, , 6 ''E' 7, 2 ''E 2, 2 '' 2, 28 'E' 2, 23 'E 28, 3 E' 23, 3 E 3, 3, 9, 7, 25 ''E,, 2, 3, 8, 9,, '' f = '' + ''E + ''E' + E + E' + '' f = '' + ''E + ''E' + 'E' + 'E () f = + E' + ''E + ''' +''E + '''E' f = + E' + ''E + ''E + ''' + '''E' 6.3 = ' + ' + ' + '' + ' = ' + ' + ' + '' + ' = ' + ' + ' + ' + '' 6.4 Prime implints: e, 'e, 'e, 'e', '', '', ''e, ''e', ''e' Essentil prime implints re unerline: = e + ''e' + 'e' + ''e + '' = e + ''e' + 'e' + ''e + 'e 43

40 6.5 ü, 3 -ü, 3, 7, 9 -- ''' 2 ü, 7 -ü, 7, 3, ü 2, 3 -ü 2, 3, 8, 9 -- '''E 32 ü 2, 8 -ü 2, 8, 3, ü 6, 7 -ü 6, 7, 8, 9 -- ''' 7 ü 6, 8 -ü 6, 8, 7, ü 6, 48 - ''E'' 48 ü 32, 48 - ''E'' 9 ü 3, 9 -ü 26 ü 7, 9 -ü 28 ü 8, 9 -ü 5 ''E 8, 26 - ''E' 29 ü 26, 3 - 'E' 3 ü 28, 29 - 'E' 39 ''E 28, 3 - '' 63 E ''E 39 ''E 63 E 6, 48 ''E'' 32, 48 ''E'' 8, 26 ''E' 26, 3 'E' 28, 29 'E' 28, 3 '', 3, 7, 9 ''' 2, 3, 8, 9 '''E 6, 7, 8, 9 ''' 6.5 () 6.5 () 6.5 () G = ''E + E + ''' + '''E +''E'' +''' +''E' G = ''E + E + ''' + '''E +''E'' +''' + 'E' Essentil prime implints re unerline in 6.5 (). If there were no on't res, prime implints 5, (26, 3), (28, 29), n (28, 3) re omitte. There is only one minimum solution. Sme s (), eept elete the seon eqution. 6.6 () Prime implints: 'E, '', 'E'', 'E, 'E'', E, 'E, ''E', ''E, ''E', ''E', ''E' G = E + 'E'' + 'E'' + 'E + ''E' + 'E + 'E G = E + 'E'' + 'E'' + 'E + ''E' + 'E + ''E G = E + 'E'' + 'E'' + 'E + ''E' + '' + 'E G = E + 'E'' + 'E'' + 'E + ''E' + '' + ''E 44

41 6.6 () Essentil prime implints re unerline in 6.6(). 6.7 Prime implints:, ',,,, ' Minimum solutions: (' + ); (' + ); ( + ); ( + ); ( + ') 6.6 () If there re no on t res, the prime implints re: 'E, '', 'E'', 'E, 'E'', E, 'E, ''E' G = E + 'E'' + 'E + 'E'' +'E + ''E' + '' G = E + 'E'' + 'E + 'E'' +'E + ''E' + 'E 6.8 () E 6.8 () G E E E = ' + ' + '' + E (''' + ') MS MS Z = ' + '' + E (' + ') + (') + G ('') MS MS MS 2 MS () Eh minterm of the four vriles,,, epns to two minterms of the five vriles,,,, E. or emple, m 4 (,,,) = ''' = '''E' + '''E = m 8 (,,,,E) + m 9 (,,,,E) 6.9 () Prime implints: ''', ', ', ''E, E, E, ''E = ''' + ' + ' + ''E + E = ''' + ' + ' + ''E + E E = ''' + ' + ' + ''E + E = ''' + ' + ' + ''E + E 6.2 E * * This squre ontins +, whih reues to. G = 'E' + E + ('') + () MS MS MS 2 45

42 7. () f Unit 7 Prolem Solutions f f = '' + '' + ' + ' ' Sum of prouts solution requires 5 gtes, 6 inputs f = ('+ ') ( + ) ( + + ') ( + '+ ') f = ('+ ') ( + ) ( + '+ ') ('+ + ') f = ('+ ') ( + ) ( + + ') ('+ '+ ') f = ('+ ') ( + ) ('+ + ') ('+ '+ ') Prout of sums solution requires 5 gtes, 4 inputs, so prout of sums solution is minimum. 7. () eginning with the minimum sum of prouts solution, we n get f = ' ( + ') + ' (' + ') 5 gtes, 2 inputs So sum of prouts solution is minimum. 2 eginning with minimum prout of sums solution, we n get f = ( + ) (' + ') (' + ' + ')) gtes, 4 inputs () ' + E' + E' + ' + ''E' = E' ( + ) + ''E' + ' ( + ) = ( + ) (E' + ') + ''E' levels, 6 gtes, 3 inputs 7.2 () E + E + E + G + G + G = E + G + E ( + ) + G ( + ) = (E + G) [ + ( + )] levels, 6 gtes, 2 inputs

43 7.3 (,,, )n = ' + ' or (' + ') = ( + ) (' + ') You n otin this eqution in the prout of sums form using Krnugh mp, s shown elow: ' ' N-OR ' ' NN-NN OR-NN ' ' ' ' NOR-OR ='+' (')'=[(')'(')']' (')'=[(+'+')('++')]' (')'=(+'+')'+('++')' (')'=[('+')']' ' ' ' ' ' ' OR-N ' ' NOR-NOR ' ' =(+)('+') (')'=['+(+)'+('+')']' (')'=['+''+]' (')'=('')'()' (')'=[[(+)('+'))']' ' ' N-NOR ' ' NN-N = ' + ' = ( + ) ( ) ('+ ') 7.4 (,,, ) = m(5,,, 2, 3) = ' + ' + ' = ' ( + ) + ' = ' ( + ) + ' gtes, inputs = ' + ' + ' ' ' ' ' 47

44 7.5 Z = ( + + ) ('+ ') ('+ ') ('+ ') Z = ( + + ) (' + ''') gtes, inputs ' 2 3 Z 7.6 Z = + + '' = ( + ) + '' ' ' ' Z 7.7 Z = E + E + E = E ( + + ) = E [ + ( + )] ' ' ' E' Z 7.8 or the solution to 7.8, see L P = ' + + '' 6 gtes 2 = '' + '' + ' 7. f (,,, ) = m(3, 4, 6, 9, ) f 2 (,,, ) = m(2, 4, 8,,, 2) f 3 (,,, ) = m(3, 6, 7,, ) = ' + ' + ' ' 2 = ' + ' ' + '' + '' 2 = ' + ' ' + '' + '' gtes 48 3 = ' + ' + '

45 7. = ( + ) ( + ') ('+'+)('++') 8 gtes 2 2 = ('++)('+'+)(+')('++') 2 = (+'+)('+'+)(+')('++') = (++)('+) 2 = (++)('++)('+) 9 gtes 3 = ('++)(+)(+') 7.3 () Using = (')' from Equtions (7-23()), p. 94: f = [(')' ()' ('')' (')']'; f 2 = [' (')']'; f 3 = [()' ('')' ()']' ' ' ' ' f ' f 2 ' ' ' f () Using = (')' from Equtions erive in prolem 7.2: f = [( + + )' + (' + )']' f 2 = [( + + )' + (' + + )' + (' + )']' f 3 = [(' + + )' + ( + )' + ( + ')']' ' f ' ' f 2 ' ' f 3 49

46 7.4 () f = ( + + ) ( + + ') ('+ '+ ') ('+ '+ ') 5 gtes, 6 inputs 7.4 () eginning with the sum of prouts solution, we get f = ' + ' + ' (' + ') = ' + ' + ' (' + ') ( + ) 6 gtes, 4 inputs ut, eginning with the prout of sums solution ove, we get f = ( + + ') (' + ' + '') 5 gtes, 2 inputs, whih is minimum n f = ' + ' + '' + '' f = ' + ' + '' + '' (two other minimun solutions) 5 gtes, 4 inputs miniml ' ' ' ' ' ' ' ' ' ' ' 7.5 () rom K-mps: = ' + ' + ' 4 gtes, inputs = ( + + ) ( + ) (' + ') 4 gtes, inputs, miniml 7.5 () rom K-mps: = + + '' 4 gtes, 9 inputs = (' + ) ( + ' + ) 3 gtes, 7 inputs, miniml ' ' ' ' 7.5 () rom K-mps: = + '' + = + '' + ' 4 gtes, inputs = ( + ) ( ' + ) ( + + ') 4 gtes, inputs, miniml ' ' 7.5 () rom K-mps: = ' + + ' 4 gtes, 9 inputs, miniml = ( + ) (' + + ') (' + + ) = ( + ) (' + + ') ( + + ) 4 gtes, inputs ' ' 5

47 7.6 () In this se, multi-level iruits o not improve the solution. rom K-mps: = ' + + ' + '' 5 gtes, 6 inputs, miniml = (' + + ) ( + + ) (' + ' + ) ( + + ') 5 gtes, 6 inputs, lso miniml Either nswer is orret. ' ' ' ' 7.6 () Too mny vriles to use K-mp; use lger. E y onsensus, then use + Y = E + E + ' + EG + E + E G E ' = E + ' + EG + E = E ( + G) + (' + E) gtes, 3 inputs, miniml E () = M(,, 2, 4, 8) () = ( + + ) ( + + ) ( + + ) ( + + ) = ( + + )( + + ) or = ( + + )( + + ) or = ( + + )( + + ) This solution hs 5 gtes, 2 inputs. eginning with the sum of prouts requires 6 gtes. 5

48 7.8 () (w,, y, z) = ( + y' + z) (' + y + z) w y' z ' y z OR-N w y' z ' y z NOR-NOR w' ' y z' y' z' N-NOR w' ' y z' y' z' NN-N w rom Krnugh mp: = wy + w'y' + wz w y w ' y' w z N-OR w y w ' y' w z NN-NN w' ' y' w' y w' z' OR-NN w' ' y' w' y w' z' NOR-OR 7.8 () (,,, ) = m(4, 5, 8, 9, 3) rom Kmp: = '' + '' + ' = '' + '' + ' = ' ( + ) (' + ' + ) ' ' ' ' ' N-OR ' ' ' ' ' NN-NN ' ' ' ' OR-NN ' ' ' ' NOR-OR ' ' OR-N ' ' ' NOR-NOR ' ' ' N-NOR ' ' ' NN-N ' 52

49 7.9 () y z = (y'+ z ) ('+ y + z') rom Kmp: = (y' + z) (' + y + z') y z' y' z f y z' y' z f 7.9 () y z ' z ( ) or ' y' y' z' f ' z ( ) or ' y' y' z' f = y z + y'z' + 'y' y z y z = y z + y'z' + 'z 7.2 () Using OR n NOR gtes: ' ' f ' ' f = '' () Using NOR gtes only: ' ' ' f ' ' ' f = ('+ ) ('+ ) ('+ ) ('+ ) ' ' 53

50 7.2 () NN gtes: = ' + ' + (Refer to prolem 5.4 for K-mp) NOR gtes: = ( + ' + ') ( + + ') 7.2 () NN gtes: = '' + '' + (Refer to prolem 5.8() for K-mp) NOR gtes: = ( + ) (' + ') (' + ) (' + ') = ( + ) (' + ') (' + ) (' + ) 7.2 (e) NN gtes: =''+'E'+E+'''+''E+'E' =''+'E'+E+''E'+'+''E =''+'E'+E+'E'+''E'+''E =''+'E'+E+'E'+''E+'E (Refer to prolem 5.9() for K-mp) NOR gtes: = (' + ' + + E) ( + ' + E') (' + ' + E) ( + + ' + E) ( + + ) (' + + E') 7.2 (g) NN gtes: f = 'y' + wy' + w'z+ wz' f = 'y' + wy' + w' + w'z f = 'y' + wy' + y'z+ wz' (Refer to prolem 5.22() for K-mp) NOR gtes: f = (w + ' + z) (w + y' + z) (w' + y' + z') f = (w + ' + z) (w + y' + z) (w' + ' + y') 7.2 () NN gtes: = '' + ' + '' (Refer to prolem 5.8() for K-mp) NOR gtes: = (' + ') (' + ') ( + + ) (' + + ) 7.2 () NN gtes: = '' + ''E + 'E + 'E + E' + '' + ''E' + '' = '' + ''E + 'E + 'E + E' + '' + ''E' + ''E' (Refer to prolem 5.9() for K-mp) NOR gtes: = (' + + E) ( + ' + ) (' + + ' + ') (' + ' + + E ) ( + ' + ' + E) (' E') (' + ' + ' + E') 7.2 (f) NN gtes: = ' + ' + ' + '' (Refer to prolem 5.22() for K-mp) NOR gtes: = ( + + ) (' + ' + ) (' + ') 7.22 () 7.22 () f = (' + ' + e) ( + ' + e') ( + + ) E ' e' e ' ' ' ' f = ( + ' + ) ( + ' + e') (' + ' + e) ( + + ) ( + ' + ') 54

51 7.23 ' ' ' ' ' ' f f = (' + ) (' + + ) ( + ') = ( + ') [' + ( + )] = ( + ') (' + + ) 7.24 () Z = e'f + 'e'f + 'e'f + gh = e'f ( + ' + ') + gh 7.24 () Z= (' + +e + f)(' + ' + )(' + ' + )(g+h) = [' + + '' (e + f)] (g + h) g h e' f z e f g h ' z 7.25 = e' + '' + = ( + ') (' + e') + = ( + ' + ) (' + + e') ' ' ' e ' lternte: = (' + + ) (' + + e') 7.26 f = 'yz + vy'w' + vy'z' = 'yz + vy' (z' + w') w z ' y z v y' f 7.27 () 7.27 () = ' + ' + ''' + ' = ' + ' + ''' + ' rw N-OR iruit n reple ll gtes with NNs. = ( + + ') ('+ ) ('+ ) ('+ '+ ') rw OR-N iruit n reple ll gtes with NORs. 55

52 7.27 () = (' + ') + ' ('' + ) ' ' ' ' ' lterntive: = ' ('' + ) + (' + ') = (' + ') + ' ('' + ) = ' ('' + ) + (' + ') = (' + ') + ' ('' + ) 7.28 () 7.28 () = + '' + '' + '' + '' + '''' = ( + + ') ( + ' + ) ( + ' + ) ( + ' + ') (' + + ) (' + + ') (' + + ') (' + ' + ) 7.28 () Mny solutions eist. Here is one, rwn with lternte gte symols. = ' (''' + ' + ') + ('' + '' + ) = ' ('('' + ) + ') + (('' + ) + '') ' ' ' ' ' ' ' 56

53 7.29 () = '' '' = ( + '') + ( + '') Mny NOR solutions eist. Here is one. = ( + ) (' + + ) ( + + ') ( + ' + ' + ) = ( + ) [ + ( + ') (' + ' + )] (' + + ) = ( + ) [ ( + ) + ' ( + ') (' + ' + )] = ( + ) [ ( + ) + ' ((' + ) + '')] ' ' ' 7.29 () = ' + + '' + ' = ( + ') + ('' + ') = ( + ') + [(' + ') ( + ')] = ( + ) ( + ) ('+ + ') ('+ + ') ' ' ' ' ' ' ' ' ' ' = (' + + ') (' + + ') ( + ) ( + ) = ( + (' + ')) ( + (' + ')) 7.3 = m(,, 2, 3, 4, 5, 7, 9,, 3, 4, 5) = + '' + '' + = + ' (' + ') + lternte solution: = + (' + ) ( + ' + ') = '' + ''

54 7.3 () 7.3 () ' ' ' ' ' ' 7.3 () ' ' ' ' ' ' ' ' ' ' 7.3 Z = [' + + E(' + GH)] G H ' E ' G' H' ' E' ' ' 58

55 7.32 f f2 f3 f = '' + f 2 = '' + '' 8 gtes f 3 = + + '' 7.33 = ' + ' + '' 6 gtes = '' + ' + '' 7.34 f y z f2 y z f3 y z f = 'y z + ' y z' + y' f 2 = y' z + 'y z + y z' 8 gtes f 3 = y' + y'z + 'y z' + y z' 7.35 () f f2 f = (' + + ) (' + ' + ') (' + ) 6 gtes f 2 = (' + + ) (' + ' + ') ( + ') 59

56 7.35 () f irle 's to get sum-of-prouts epressions: f = ' + ''' + ' 6 gtes f 2 = ' + ''' + ' Then onvert iretly to NN gtes. f () irle s 2 f = ( + + ) ('+ ') ('+ ) 7 gtes f 2 = ( + + ) (' + ) (' + ' + ') 7.36 () irle 's to get sum-of-prouts epressions: 2 Then onvert iretly to NN gtes f = ' + ' + ' 7 gtes f 2 = ' + ' + ' 6

57 7.37 () f = + ' + ' 2 f 2 = ' + ' + ' ' ' ' f f () f = ( + ) ( + ) ( + ) ( + + ') 2 f 2 = ( + ) ( + ) ( + + ') ('+'+') ' f f 2 ' ' ' 7.38 () f f = '' + ' + ' f 2 = '' + ' + ' 7.38 () f2 f = ('+ ) ('+ ') ( + ') ( + '+ ') f 2 = ('+ ) ('+ ') ( + '+ ') ( + ' + ) 6

58 7.39 () The iruit onsisting of levels 2, 3, n 4 hs OR gte outputs. onvert this iruit to NN gtes in the usul wy, leving the N gtes t level unhnge. The result is: ' ' e f ' g h () One solution woul e to reple the two N gtes in () with NN gtes, n then inverters t the output. However, the following solution vois ing inverters t the outputs: = [( + ') + ] (e' + f) = e' + 'e' + e' + f + 'f + f = e' ( + ') + (e' + f) + f ( + ') 2 = [( + ') + g'] (e' + f) h = h (e' + 'e' + f + 'f) + g'h (e' + f) = h [e' ( + ') + f ( + ')] + g'h (e' + f) ' e' e f ' f g' h h 2 62

59 Unit 8 Prolem Solutions 8. W Y V Z t (ns) 8.2 () = ''' + + ' 8.2 () (ont) Stti -hzrs re: n = ( + ') ('+ + ) ( + + ') Stti -hzrs re: n 8.2 () 8.2 () t = ( + ') (' + + ) ( + + ') (' + + ) ( + + ') 8.3 () t = ''' + + ' + '' + E G t (ns) Glith (stti '' hzr) 8.3 () Moifie iruit (to voi hzrs) 8.4 G = '' + + ' = ; = Z; = Z = ; = + Z = ; E = ' = ; = ' = ; G = = ; H = + = See L Tle 8-, P = =, = = So = ' + '' + = ut in the figure, gte 4 outputs =, initing something is wrong. or the lst NN gte, = only when ll its inputs re. ut the output of gte 3 is. Therefore, gte 4 is working properly, ut gte 3 is onnete inorretly or is mlfuntioning. 63

60 W Y V Z t (ns) = Z; = ; = Z' = ; = Z = ; E = Z; = + + = ; G = ( Z)' = ' = ; H = ( + )' = ' = = = =, so = ( + ' + ') (' + + ') (' + ' + ) = ut, in the figure, gte 4 outputs =, initing something is wrong. or the lst NOR gte, = only when ll its inputs re. ut the output of gte is. Therefore, gte 4 is working properly, ut gte is onnete inorretly or is mlfuntioning. 8.7 Z = ' + '' + ' ' Stti -hzrs lie etween n Without hzrs: Z t = '' + ' + '' + '' + '' 8. () (,,, ) = m(, 2, 5, 6, 7, 8, 9, 2, 3, 5) There re 3 ifferent minimum N-OR solutions to this prolem. The prolem sks for ny two of these. 8. () = + ' + ' ' + ''' Solution : -hzrs re etween n = ( + + ') ( + ' + + ) (' + ' + ) (' + + ') -hzr is etween Either wy, without hzr: t = ( + + ') ( + ' + + ) (' + ' + ) ( + ' + ') (' + + ') = + ' + ''' + ' Solution 2: -hzrs re etween n 64 = + ' + ''' + ' ' Solution 3: -hzrs re etween n Without hzrs: t = + ' + ''' + '' + ''' + ' = ( + + ') ( + ' + + ) (' + ' + ) ( + ' + ') -hzr is etween

61 Unit 9 Prolem Solutions 9. See L p. 636 for solution. 9.2 See L p. 636 for solution. 9.3 See L p. 637 for solution. 9.4 See L p. 637 n igure 4-4 on L p y y y 2 y 3 y y y y 2 3 y y y y 2 3 y y y y 2 3 = y 3 + y 2 = y 3 + y 2' y = y 3 + y 2+ y + y 9.6 See L p. 638 for solution. 9.7 See L p. 638 for solution. 9.8 See L p for solution. 9.9 The equtions erive from Tle 4-6 on L p. re: 9. Note: 6 = 4 ' n 5 = 4. Equtions for 4 through n e foun using Krnugh mps. See L p for nswers. = 'y' in + 'y in ' + y' in ' + y in out = ' in + 'y + y in See L p. 639 for PL igrm. 9. () = '' + ' + ' Use 3 N gtes ' = ['' + ' + ']' = [' ( +') + ']' = [( + + ') (' + ')]' = '' + Use 2 N gtes 9. () = '' + '' Use 2 N gtes ' = ('' + '')' = [(' +') (' + ') (' + ') (' + ')]' = Use 4 N gtes 9.2 () See L p. 64, use the nswer for 9.2 (), ut leve off ll onnetions to n '. 9.2 () See L p. 64 for solution. 9.3 Using Shnnon s epnsion theorem: = 'e' + ''e + ''e + 'e' = ' (e' + ''e + 'e') + (''e + ''e + 'e') = ' [e' ( + ') + ''e] + [(' + ') 'e + 'e'] = ' (e' + ''e) + (''e + ''e + 'e') The sme result n e otine y splitting Krnugh mp, s shown to the right. E = = 65

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