Digital Circuit Engineering

Transcription

1 igitl ircuit Engineering ^ ' fctoring IGITL st istributive X + X = X( + ) 2nd istributive (X + )(X + ) = X + (X + )(X + )(X + ) = X + Swp (X + )(X + ) = X + X VLSI ESIGN The Most ommon Stupid Errors Using Not Using X.Y = XY YX + X = X X + = X Y + XY = X + Y Generl emorgn F(, b,... z,+,.,,0) F(, b,... z,.,+,0,) rleton University 2006 dig4fctoringe.fm p. 26 Revised; Jnury 25, 2006 Slide i Multiplying Out nd Fctoring Sum-of-Products, Product-of-Sums Multiplying Out Use Use 2 nd Swp, then Use Krnugh Mp Fctoring Use 2 Tke the dul Þ multiply out Þ tke the dul bck Find F using emorgn, then use Krnugh mp rleton University dig4fctoringe.fm p. 27, Revised; Jnury 25, 2006 omment on Slide

2 Stndrd (nonicl) Forms Sum-of-Products Product-of-Sums These re stndrd templtes or forms Every logicl expression cn be converted to either of these forms. Sum of Products Σ of Π, OR of Ns bc + cde + d + f + db Single vribles Ned together into terms. These terms re ORed together. Inversions re only over individul vribles. No brckets. Product of Sums. Π of Σ, N of ORs Single vribles ORed together into terms. These terms re Ned together. Inversions re only over individul vribles. rckets only round vribles in OR terms Questions Is b + cd + b(d+e) Σ of Π? Is ( +bc)(d+e) Π of Σ? (+b+c)(c+d+e)(+d)(f)(d+b) dig4fctoringe.fm p. 28 Revised; Jnury 25, 2006 Slide 4 Stndrd (nonicl) Forms Sum-of-Products Sum-of-Products These must be expressed lgebriclly s the OR of Ns. One ws not llowed to put in brckets, XORs, nor long inversion brs. Thus none of the exmples on the left below re true sum of products. NOT Σ of Π bc + b(+d) + de bc + bcd +bf (bc + bfg + bcd) c + bg d+bc bc These re Σ of Π fter finding n lgebric Σ of Π form, it cn be esily chnged to NN-NN logicfor implementtion. Product of Sums These must be expressed lgebriclly s the N of ORs. One must use single-letter terms ORed together, with no XORs, nor long inversion brs. Thus none of the exmples on the left below re true product of sum. (+b+c)b(+d) (+b+c)(d+e)+d (e+b+c) (+b+c)(b+c+d)(+b) ( c+b)(g+h) +b+c These re Π of Σ NOT Π of Σ fter finding lgebric Π of Σ form, it cn be esily chnged to NOR-NOR logic for implementtion. rleton University dig4fctoringe.fm p. 29, Revised; Jnury 25, 2006 omment on Slide 4

3 Sum-of-Products Product-of-Sums Two nonicl Forms Sum of Products (Σ of Π) bc + e + ce + bd +... OR of Ns c b e e c d b Σ of Π Σ of Π (NN-NN) n be implemented s NN-NN logic NN Product of Sum (Π of Σ) ul of Σ of Π (+b+c)(+e)(+c+e)(+b+d)(... N of ORs n be implemented s NOR-NOR logic b c e e c b d c b e e c d b Π of Σ NOR c b e e c d b Π of Σ (NOR-NOR) dig4fctoringe.fm p. 30 Revised; Jnury 25, 2006 Slide Sum-of-Products Product-of-Sums nonicl Forms efinition of oolen Function function is wy of getting the output from the inputs. There re mny templtes (forms) to define oolen Function. form tht cn define ll oolen functions is clled cnonicl. Some nnonicl Forms. The truth tble is very bsic form Σ of Π expression. b + b Π of Σ expression. ( + b)( + b) (fctored form) Krnugh mps form tht will be introduced shortly. b b nonicl Forms inry decision digrms, which will be used lter to build rbitrry circuits with 2-input muxes. Why we sy Σ of Π for Sum of Product n Σ is used for repetitive ddition, s in å x i = x 0 + x + x 2 + x, n i = 0 n Π is used for repetitive product, s in x = x x x x i 0 2 n i = 0. nonicl mens ccording to the rule or lw, prticulrly the church lw. It is stndrd, logicl, nd essentilly unique wy of writing ny function. rleton University dig4fctoringe.fm p. 3, Revised; Jnury 25, 2006 omment on Slide

4 Fctoring nd Multiplying Out Trnsforming Σ of Π Π of Σ Multiplying out Trnsforms Π of Σ Σ of Π ( + c)(b + + d) b + cd Fctoring Trnsforms Σ of Π Π of Σ b + cd ( + c)(b + + d) Why Fctor? : Typiclly both form hs bout the bout the sme size, but sometimes fctoring cn sve significnt logic. b + c + d + e = (b+c+d+e) cb + c b + cd + c d = ( + c)(b + d)( + c) c + e +bc + be = ( + b)(c + e) 5 gtes, 8 letters 2 gtes, 5 letters 5 gtes, 2 letters 4 gtes, 6 letters 5 gtes, 8 letters 3 gtes, 4 letters 2: Sometimes NOR-NOR logic my be desired (Fst fll time). dig4fctoringe.fm p. 32 Revised; Jnury 25, 2006 Slide 2 Fctoring nd Multiplying Out Σ of Π Π of Σ Σ of Π Π of Σ Why fctor? Usully the (Π of Σ) nd (Σ of Π) forms re bout the sme complexity, but switching forms for prt of the circuit cn sve significnt logic. Logic minimiztion progrms will switch from one form to the other in different prts of lrge circuit. 9. PROLEM Identify the following s: Σ of Π, Π of Σ, or neither.. (W + Y + Z)(X + Y + Z)(W + X)(W + Z)(X + Y + Z)(W + X + Z) 2. (b + c)(d + ce)( + b + c + d) 3. dc + (bc)+ cb 4. bc + c(+b) + dec +b 5. + b + cd rleton University dig4fctoringe.fm p. 33, Revised; Jnury 25, 2006 omment on Slide 2

5 Multiplying Out; Methods Multiplying Out hnge Π of Σ Σ of Π ( )( )( ) ( ) + ( ) + ( ) st istributive Lw () x( + ) = x + x Methods of Multiplying Out. Use the st distributive lw x( + b) = x + b - This lwys works, but - it usully gives very very long result. 2. Use 2 nd/or Swp before using - Gives shorter nswers - Shorter steps to get nswer - Requires more thinking 3. Use emorgn s Lw to get F Plot F on K-mp Then plot F - Esiest to do - Gives the simplest Σ of Π nswer. - Very messy for 5 or more vribles. Exmple ( + )( + + ) Use () = = Exmple ( + )( + + ) = ( + )( + [ + ]) = [ + ] + = + + Exmple F=( + )( + ) Use (em) F = + Use Mp F = + Use () Use = 0 Use (Sw) (x+)(x+) = x + x F F dig4fctoringe.fm p. 34 Revised; Jnury 25, 2006 Slide 3 Multiplying Out; Methods Σ of Π Π of Σ Three methods of multiplying out Using Using mny times is the strightforwrd wy, is very esy to use. Unfortuntely if it is done utomticlly with no thinking, it cn get very long. If one uses simplifiction (X + Xy =X) nd bsorption (X + Xy = X y) t every chnce, the work is shorter, but the finl expression still my be much longer thn necessry. Using 2 nd Swp before using If the expression hs repeted letters like Xb + Xcd.. or Xb + Xdg then 2 or the Swp rule cn do preliminry consolidtion before using. In ll but few cses one must use for finl clenup. Using Krnugh mps This is the esiest method for four or five vribles, it lwys gives the smllest nswer, it esily hndles don t cres, but gets very complex for over five input vribles. It is the method of choice for most smll problems. Three methods of fctoring Using 2 This is the strightforwrd wy, unfortuntely it uses the unfmilir 2 distributive lw which mkes the lgebr hrder for most people. Using dulity nd This is lgebriclly just s difficult s the previous method. However using the more fmilir () mkes it esier for most people. Using Krnugh mps This is quite esy for four vribles, but more complex for over five input vribles. Esily hndles d s. rleton University dig4fctoringe.fm p. 35, Revised; Jnury 25, 2006 omment on Slide 3

6 Multiply Out Using st istributive Lw Method Using Step : Simplify, Simplify, Simplify, Simplify..., Use Step 2: Simplify, Use Step 3: Simplify, Use... Finl Step: Simplify Exmple ( + X)( + X)( + X) Use () = [( + X) + X( + X)]( + X) Use () = [ + X + X + X]( + X) Use (S) x + x = x = [ + X]( + X) Use () = ( + X) + ( + X)X Use () = ( + X + X + X Use (S) x + cx = x = + X st istributive Lw x( + ) = x + x Wys to Simplify (S) - X +X = X - X +X = X + () - Krnugh mp dig4fctoringe.fm p. 36 Revised; Jnury 25, 2006 Slide 4 Multiply Out Using st istributive Lw Exmple of Multiplying Out Exmple of Multiplying Out = = ( + )( + )(F + G + H) = ( )(F + G + H) ( ) F + ( ) G + ( ) H F + F + F + F + G + G + G + G + H + H + H + H ll letters re different, no simplifiction possible Use () rewrite Use () With ll the letters different, there is no wy to simplify. The expressions get long rpidly. Using () lwys works, it is esy on the brin, but hrd on the pencil. lso the simplifictions must be done by other mens. rleton University dig4fctoringe.fm p. 37, Revised; Jnury 25, 2006 omment on Slide 4

7 Multiply Out Using st istributive Lw Method Using Step : Simplify, Use Step 2: Simplify, Use Step 3: Simplify, Use... Finl Step: Simplify Exmple: ( + )( + F + )( + )( + F + ) 0 c +cy = c How to Simplify X +X = X (S) X +X = X + () Krnugh mp or concensus if you hve to Use twice = ( + F F + )( + F F + ) = (F + )(F F + ) Use 0 bf + bfy = bf = FF + F + F + FF + F + F F + ) Put letters in order for mp = F F + Use Use mp consensus (,) = F + ( + + ) + F (cons) = F + ( + ) + F () = F F st istributive Lw () x( + ) = x + x 5-vrible mp F F F F = F + + continued below dig4fctoringe.fm p. 38 Revised; Jnury 25, 2006 Slide 5 Multiply Out Using st istributive Lw Exmple of Multiplying Out Simplifying by onsesus insted of 5-Vrible Mp (continued) = F F x=x+xb = F + + F + + F () = F +F + F + + () = F( + + ) + +. The 5-vrible mp is not covered until the next section. = F( + + ) + + = F( + ) = F + + F + = F + + Exmple: Multiply Out, Not pure Π of Σ Multiply out the expression below (see next comment pge for source) F = ( + ) [( + ) + ( + )] ( + + ) Use () = { [( + ) + ( + )] + )[( + ) + ( + )]} ( + + ) xx = 0 xx = 0 = {( + ) + ( + )} ( + + ) xy + x = x = { )} ( + + ) { + + )} + + = + { + + )} = + + { + + )} + xy + x = x ollect terms = Mp shows there reno more simplifictions Use () Use () () (cons) () () x=fx+x This pseuo mtrix helps rrnge the messy multipliction rleton University dig4fctoringe.fm p. 39, Revised; Jnury 25, 2006 omment on Slide 5

8 Multiply Out Using 2 nd Swp, efore Method Using 2 nd Swp Step : Simplify Step 2: Use 2 nd/or Swp, Simplify Step 3: Use, Simplify... Repet Steps 2 nd 3 Exmple: ( + + )( + )( + + )( + )( + + ) (x)(x+y) = x = ( + + )( + )( + + )( + )( + + ) = ( + + )( + + )( + )( + ) 2 Sw ( + + ) ( + ) x +xy = x 2nd istributive Lw (2) (X+c)(X+d) = X + cd Swp (y+x)(z+x)=zx+yx lwys check for obvious simplifictions Rerrnge to use (2) nd Swp Use (2) nd Swp (X+c)(X+d) = X + cd (y+x)(z+x)=zx+yx ( + + ) = + ( + + ) = + + = + + = Use () ollect terms heck mp for further simplifictions dig4fctoringe.fm p. 40 Revised; Jnury 25, 2006 Slide 6 Multiply Out Using 2 nd Swp, efore Exmple onsolidting with 2 nd Sw Exmple onsolidting with 2 nd Sw F = ( + )( + + )( + )( + + )( + + ) = ( + )( + ) ( + + )( + + ) ( + + ) = ( + ) [( + ) + ( + )] ( + + ) This is s much s one cn do using 2 nd Sw. One must use from here on. It is reduced to the problem on the previous pge which gives F = rrnge terms to redy to use Swp Use Swp twice Use () Johnny missed simplifiction in the st line. n you see where? 0. PROLEM Multiply out. Remember to check for obvious simplifictions before strting. (W + Y + Z)(X + Y + Z)(W + X)(W + Z)(X + Y + Z)(W + X + Z) Hint: Tke the dul, simplify the dul, then tke the reverse dul.. PROLEM Multiply out to get four terms of three letters ech. The nswer should be very symmetric on n \ Krnugh mp. ( + )( + + )( + + )( + + )( + + ) Hint: Only 4 inputs? If so the next (K-mp) method is simpler. rleton University dig4fctoringe.fm p. 4, Revised; Jnury 25, 2006 omment on Slide 6

9 Multiply Out Using emorgn nd K-Mps Method Using K-Mps Step : Find F Using Generl emorgn Step 2: Plot F on K-Mp Step 3: Plot F Using the 0 Squres Step 4: Get expression for F from mp Exmple: () Find the inverse using generl emorgn F = ( + + )( + )( + + )( + ) F dul = () + ( ) + () + () F = (2) Plot inverse on K-mp (3) Plot the F on K-mp; Put s where F ws 0 s. (4) ircle F on K-mp F = + + Mp of function, F Mp of function, F Mp of inverse, F dig4fctoringe.fm p. 42 Revised; Jnury 25, 2006 Slide 7 Multiply Out Using emorgn nd K-Mps The Esy Wy to Multiply Out The Esy Wy to Multiply Out F is esy to find using emorgn s lw. F is esy to find from K-mp of F. These re the essentil fcts used for multiplying out using K-mp. When is the K-mp the best method? For most problems done by hnd is by fr the esiest wy to fctor. Six or more vribles will give problem too big for K-mp. Using lgebr my be esier if ll the vribles, or lmost ll, re different.see omment on Slide 4. lgebr my be esier for converting smll prtil expression inside long expression. 2. PROLEM: Multiply out using emorgn s Lw nd Krnugh mp, to get two terms of 2 letters nd one of 3 letters. ( + + )( + )( + + )( + )( + + ) 3. PROLEM (SOLUTION TO LTER FTORING PROLEM) Multiply out to get four terms of three letters ech. Use Krnugh mp. ( + )( + + )( + + )( + + )( + + ) rleton University dig4fctoringe.fm p. 43, Revised; Jnury 25, 2006 omment on Slide 7

10 Multiply Out: Using emorgn nd K-Mps Method Using Krnugh Mp Steps: () Given F = (Π of Σ expression) F = ( + )( + + )( + + )( + + ) (2) Invert F using emorgn s lw to get F s Σ of Π F = (3) Plot it on mp. (4) Mke mp for F, It hs where F hd 0 (5) ircle the F mp (6) Write out the eqution for F F = Mp of F Mp of F Mp of F dig4fctoringe.fm p. 44 Revised; Jnury 25, 2006 Slide 8 Multiply Out: Using emorgn nd K-Mps Multiply Out With d s Multiply Out With d s If some input combintions re never used, these become don t cre outputs. Go through the norml steps () Find F (2) Mke the mp of F from tht of F. Then identify the d s on the mp of F. Finlly circle the mp normlly to find the minimum Σ of Π expression. 4. PROLEM: Find the minimum Σ of Π expression for F using the don t cres to best dvntge. F = (W+X+Y)(W+X+Z)(X+Y+Z) The input combintions G = W XYZ + WXYZ) never hppen, so these squres re d s. Z YZ WX d 0 X W 0 d Y Helpful mp rleton University dig4fctoringe.fm p. 45, Revised; Jnury 25, 2006 omment on Slide 8

11 Fctoring Fctoring, the ul of Multiplying Out hnge Σ of Π Π of Σ Exmple ( ) + ( ) + ( ) ( )( )( ) Methods of Fctoring. Use the 2nd distributive lw x + b = (x + )(x + b) - lwys works, but - very long nd slow. 2. Tke the dul Use multiply-out method Tke the dul bck - hnges unfmilir 2 to fmilr - In theory the sme mount of work, but esier to grsp. + + = (+)(+) + = [(+)(+) + ][(+)(+) + ] Exmple F UL = ( + )( + ) = ( + ) = + Tke ul Use (2) Use (2) Use (2) = [(+) + ][(+) + ][(+) + ][(+) + ] Use (2) = [ + ][++ ][++ ][+ + ] nd += = [ + ][+ + ] Use x(x+y) = x F = + + (ul) 2 = F = (+)( + + ) Use (2) Use () Tke ul ck 3. Plot F on K-mp Plot F using the 0 squres Find F using emorgn - Esiest to do. - Gives the simplest Π of Σ nswer. - Very messy for 5 vribles or more. Exmple F = + Plot on Mp Get F from Mp F = + Use (em) F=( + )( + ) F F dig4fctoringe.fm p. 46 Revised; Jnury 25, 2006 Slide 9 Fctoring Four methods of fctoring Four methods of fctoring Using 2 Using 2 mny times is the brute force wy. Unfortuntely students find 2 hrd to use, nd the expnsion my get very long. It helps to use simplifiction X (X+y) = X nd bsorption X (X+y) = Xy t every chnce, but these rules re lso more difficult thn their dul rules. If ll the letters re different, then ll one cn use is 2. Using nd Swp before using 2 This is the dul of method given for multiplying out. Unfortuntely it uses the unfmilir rules extensively, nd we suggest using dulity.this method is not even mentioned on the min slide. Using dulity nd This is lgebriclly just s difficult s the previous method, but the more fmilir rules mkes it seem esier. Using Krnugh mps This is the esiest method for four or five inputs, it lwys gives the smllest nswer, it esily hndles don t cres, but it gets very messy for over five inputs. It is the method of choice for most smll problems. Three methods of multiplying out (compre) Using Using mny times is the strightforwrd wy, is very esy to use. Unfortuntely the result cn get very long. Using simplifiction (X + Xy =X) nd bsorption (X + Xy = X y) frequently will help. Using 2 nd Swp before using If the expression hs repeted letters like Xb + Xcd.. or Xb + Xdg then 2 or the Swp rule cn do preliminry consolidtion before using. In ll but few cses one must use for finl clenup. Using Krnugh mps This is quite esy for four vribles, but more complex for over five input vribles. Esily hndles d s rleton University dig4fctoringe.fm p. 47, Revised; Jnury 25, 2006 omment on Slide 9

12 Fctoring Using 2 Method Using 2 Step : Simplify, Use 2 Repet: Step until done... Unless the problem is very simple the other methods will be esier. 2nd istributive Lw (2) X + cd = (X+c)(X+d) Exmple + X Use (2) ( + X) ( + X) Use (2) gin ( + X)( + X)( + X) Get extended (2) + X = ( + X)( + X)( + X) Exmple + Use (2) gin ( + ) ( + ) Use (2) gin, twice ( + ) ( + )( + ) ( + ) dig4fctoringe.fm p. 48 Revised; Jnury 25, 2006 Slide 0 Fctoring Using 2 Fctoring Using 2 Fctoring Using 2 The expression proven on the slide is: The extended (2) + X = ( + X)( + X)( + X) The dul is the extended () ( + + )X = X + X + X Exmple + ( + ) ( + ) ( + ) ( + )( + ) ( + ) ( + )( + ) ( + ) = 5. PROLEM Fctor + + rleton University dig4fctoringe.fm p. 49, Revised; Jnury 25, 2006 omment on Slide 0

13 Fctoring Using ulity Method 2: Fctoring Using ulity Step. Tke the dul to get fctored form ut not the right one Step 2. Multiply out the dul to get sum-of products. The right one Step 3. Tke the dul bck to get the fctored form. Fctoring Using ulity The expression to fctor is Σ of Π F = + + () Tke its dul to get Π of Σ. F UL = ( + )( + )( + + ) (2) Multiply out the dul to get Σ of Π gin F UL = This dul identity is ( + )( + )( + + ) = F UL = (3) Tking the dul bck gives vlid identity with the desired Π of Σ. + + = F = ( + )( + )( + )( + ) Multiply Out etils ( + )( + )( + + ) = ( + )( + + ) = = (2) () cx+c=c Multiplying out is bsed on (). Esier for people, thn fctoring bsed on (2). lgebr of one is the dul of the lgebr of the other dig4fctoringe.fm p. 50 Revised; Jnury 25, 2006 Slide Fctoring Using ulity hnging Fctoring into Multiplying Out hnging Fctoring into Multiplying Out Fctoring is overted to Multiplying Out, its ul Problem We tke fctoring problem which is confusing, becuse fctoring is bsed on (2). This lw is not norml lgebric lw nd is hrder to work with. In the dul spce, the dul expression is lredy fctored. The problem is trnsformed into multiplying out, which is bsed on the first distributive lw (). () is more fmilir, nd hence multiplying out is usully esier thn fctoring. Multiplying out in the dul spce does not give the nswer. One tke the dul of the nswer. This will then be the fctored form of the originl expression. 6. PROLEM Show tht F = tkes only 8 letters or 2 gte inputs in fctored form. rleton University dig4fctoringe.fm p. 5, Revised; Jnury 25, 2006 omment on Slide

14 Fctoring Using ulity Fctoring Using ulity Exmple: equl F = Minus 25% if you sy these re equl Tke dul F UL = ( + + )( + + )( + + )( + ) Rerrnge to use (2) = ( + + )( + + )( + + )( + ) Use (2) = [ + ( + )( + )] [ + ( + )] Use Swp = ( + )) + ( + )( + ) equl Use Swp = ( + )) + ( + ) Use () = = OK heck on mp Tke dul F = ( + + )( + + )( + )( + + ) dig4fctoringe.fm p. 52 Revised; Jnury 25, 2006 Slide 2 Fctoring Using ulity Fctoring In the ul Spce Fctoring In the ul Spce Exmple; Fctor Tke the dul dul = ( + + )( + + )( + + )( + + ) Note the excess of nd rerrnge for (2) = ( + + )( + + )( + + )( + + ) Use (2) = [ + ( + )( + )] [( + ( + )( + )] Use Swp = ( + )( + ) + ( + )( + ) Use Swp = ( + ) + ( + ) Use () = Mp shows no Tke the dul more simplifictions (dul) 2 = fctored form = ( + + )( + )( + + )( + + ) 7. PROLEM Fctor EF + E + E + EF rleton University dig4fctoringe.fm p. 53, Revised; Jnury 25, 2006 omment on Slide 2

15 Fctoring Krnugh Mps nd emorgn Method 3: Fctoring Using K-Mp Step. Plot function F on K-mp Step 2. Plot F by interchnging 0 on the mp. Step 3. ircle the mp to get F. Step 4. Write out the expression for F. Step 5. Use emorgn to get bck F in fctored form. Exmple: Given F = ( Σ of Π expression) F = () Plot it on mp. (2) Mke mp for F, It hs where F hd 0 (3) ircle the F mp (4) Write out the eqution for F F = (5) Invert F using emorgn s lw to get F s Π of Σ F = ( + )( + + )( + + )( + + ) Mp of F Mp of F Mp of F dig4fctoringe.fm p. 54 Revised; Jnury 25, 2006 Slide 3 Fctoring Krnugh Mps nd emorgn Method 3: Fctoring Using Krnugh Method 3: Fctoring Using Krnugh Mps This method is probbly the esiest, nd lest error prone, for up to four vribles. Five vribles is t lest twice the work of four. bove 5 it gets very messy. It is very esy to incorporte don t cres with this method. 8. PROLEM Fctor EF + E + E + EF Using Krnugh mp nd compre your nswer with the previous problem if you did it. 9. PROLEM Fctor Use Krnugh mp nd obtin the minimum Π of Σ expression. 20. PROLEM Show tht F = tkes only 8 letters or 2 gte inputs in fctored form nd compre with Problem 6. rleton University dig4fctoringe.fm p. 55, Revised; Jnury 25, 2006 omment on Slide 3

16 Fctoring Using K-Mps nd emorgn Fctoring Using Krnugh Mps Steps: Given F = ( Σ of Π expression) F = Not the minimum but it doesn t mtter () Plot it on mp. (2) Mke mp for F, It hs where F hd 0 (3) ircle the F mp (4) Write out the eqution for F F = + + (5) Invert F using emorgn s lw to get F s Π of Σ F = ( + )( + + )( + ) Mp of F Mp of F Mp of F dig4fctoringe.fm p. 56 Revised; Jnury 25, 2006 Slide 4 Fctoring Using K-Mps nd emorgn Method 3: Fctoring Using Krnugh Exmple: fctoring without using mp! This uses the esiest lgebr, dulity nd the swp rule, for fctoring Fctor E dul = ( + )( + + )( + )( + + )( + + E) = = ( + )( + ) ( + + ) ( + + E) ( + + ) = ( + ) [( + E) + ( + )] ( + + ) = { [( + E) + ( + )] + )[( + E) + ( + )]} ( + + ) = {( + ) + ( + E)} ( + + ) = { E)} ( + + ) Use () { E)} { E)} = + + E + { E)} + E ollect terms dul = E + E redundnt Tke dul bck get (dul) 2 = fctored form = (++)(++)(+++)(+++E) xy + x = x Tke dul rrnge terms to redy to use Swp Use Swp twice Use () Use () (x + )(x +b) = xb + x + E + E (x+xe=x) = + E + E + E () = + E( + + ) (consensus) = + E( + ) = + E () (x+xe=x) rleton University dig4fctoringe.fm p. 57, Revised; Jnury 25, 2006 omment on Slide 4

17 ommon Errors on t sy (+b+c)b(+d) is lredy Π of Σ lwys simplify, look for x+xz = x nd x + xy = x + Y before you strt using or 2. It sves trees. dig4fctoringe.fm p. 58 Revised; Jnury 25, 2006 Slide 5 Fctoring Using K-Mps nd emorgn Method 3: Fctoring Using Krnugh rleton University dig4fctoringe.fm p. 59, Revised; Jnury 25, 2006 omment on Slide 5

18 dig4fctoringe.fm p. 60 Revised; Jnury 25, 2006 Method 3: Fctoring Using Krnugh rleton University dig4fctoringe.fm p. 6, Revised; Jnury 25, 2006