Multiply Out Using DeMorgan and K-Maps

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1 Multiply Out Using emorgan and K-Maps Method III: Using K-Maps Step : Find F using General emorgan Step 2: Plot F on a K-Map Step 3: Plot F using the 0 Squares Step 4: Get expression for F from map Example: () Find the inverse using general emorgan F = ( + + )( + )( + + )( + ) F dual = () + ( ) + () + () F = (2) Plot F on a K-map (3) Plot the F on a K-map; Put s where F was 0 s. (4) ircle F on K-map F = + + Map of function, F Map of function, F Map of inverse, F dig4factoringh.fm p. 44 Revised; January 28, 2009 Slide 9 Multiply Out Using emorgan and K-Maps The Easy Way to Multiply Out The Easy Way to Multiply Out F is easy to find using emorgan s law. F is easy to find from a K-map of F. The above are the two essential facts used for multiplying out using a K-map. When is the K-map the best method? For most problems done by hand is by far the easiest way to factor. Six or more variables will give a problem too big for a K-map. Using algebra may be easier if all the variables, or almost all, are different.see omment on Slide 5. lgebra may be easier for converting small partial expression inside a long expression PROLEM: Multiply out using emorgan s Law and a Karnaugh map, to get two terms of 2 letters and one of 3 letters. ( + + )( + )( + + )( + )( + + ) 4-5. PROLEM (Solution to a later factoring problem,and an earlier algebraic problem) Multiply out to get an expression with eight letters. Use a Karnaugh map. ( + )( + + )( + + )( + + )( + + ) 4-6. PROLEM (Remember Simplify, simplify, simplify!) multiply out problem using,,,,e and F as variables. ( + )( + + )(E + + )( + + )(E + + )( + F) arleton University dig4factoringh.fm p. 45, Revised; January 28, 2009 omment on Slide 9

2 Multiply Out: Using emorgan and K-Maps Example Using a Karnaugh Map Steps: () Given F = (Π of Σ expression) F = ( + )( + + )( + + )( + + ) (2) Invert F using emorgan s law to get F as Σ of Π F = (3) Plot it on a map. (4) Make a map for F, It has where F had 0 (5) ircle the F map (6) Write out the equation for F F = dig4factoringh.fm p. 46 Revised; January 28, 2009 Slide 0 Multiply Out: Using emorgan and K-Maps Multiply Out With d s Multiply Out With d s If some input combinations are never used, these become don t care outputs. o the normal steps up through finding F () Make the map for F (2) Make the map for F from that of F. Then identify the d s on the map of F. Finally circle the map normally to find the minimum Σ of Π expression PROLEM: Find the minimum Σ of Π expression for F using the don t cares to best advantage. F = (W+X+Y)(W+X+Z)(X+Y+Z) The input combinations W XYZ, and WXYZ never happen, so these map squares are d s. Z YZ WX d 0 X W 0 d Y Helpful map 4-8. PROLEM: Find the minimum Σ of Π expression for G using the don t cares to best advantage. G = ( + )( + + )( + + )( + + ) The input combinations, and never happen. arleton University dig4factoringh.fm p. 47, Revised; January 28, 2009 omment on Slide 0

3 Factoring Factoring, the ual of Multiplying Out hange Σ of Π Π of Σ Example I ( ) + ( ) + ( ) ( )( )( ) Methods of Factoring I. Use the 2nd distributive law x + ab = (x + a)(x + b) - lways works, but - very long and slow. II. Take the dual Use a multiply-out method Take the dual back - hanges unfamiliar 2 to familar - In theory the same amount of work, but easier to grasp. III. Plot F on a K-map Plot F using the 0 squares Find F using emorgan - Easiest to do. - Gives the simplest Π of Σ answer. - Very messy for 5 variables or more. + + Use (2) = (+)(+) + Use (2) = [(+)(+) + ][(+)(+) + ] Use (2) = [(+) + ][(+) + ][(+) + ][(+) + ] Use (2) = [ + ][++ ][++ ][+ + ] and a+a=a = [ + ][+ + ] Use x(x+y) = x Example II F = + + F = ( + )+ F = {( + )}+ F UL = {+ )} F UL = + (ual) 2 = F = (+)( + + ) Example III F = + Plot on Map Get F from Map F = + Use (em) F=( + )( + ) Use () Take ual Use () Take ual ack F F dig4factoringh.fm p. 48 Revised; January 28, 2009 Slide Factoring Four methods of factoring Four methods of factoring I) Using 2 Using 2 many times is the brute force way. Unfortunately students find 2 hard to use, and the expansion may get very long. It helps to use simplification X (X+y) = X and absorption X (X+y) = Xy at every chance, but these rules are also more difficult than their dual rules. If all the letters are different, then all one can use is 2. II) Using duality and This in theory, is just as difficult as the previous method, but the more familiar rules makes it seem easier. IIa) Using a bit, then use duality and, as in II) above can do some factoring and often helps at the start. Example I above, using before 2, can be done in two lines. ++ =(+) + (using first) ={+}{+(+)} (using 2) III) Using Karnaugh maps This is the easiest method for four or five inputs, it always gives the smallest answer, it easily handles don t cares, but it gets very messy for over five inputs. It is the method of choice for most small problems. Three methods of multiplying out (compare with factoring) Using Using many times is the straightforward way, is very easy to use. Unfortunately the result can get very long. Using simplification (X + Xy =X) and absorption (X + Xy = X y) frequently will help. Using 2 (or equ) and Swap before using Use 2 or the equivalent (take the dual and use ) and the Swap rule can do initial consolidation before using. In most cases one must use for final cleanup. Using Karnaugh maps This is quite easy for four variables, but more complex for over five input variables. Easily handles d s arleton University dig4factoringh.fm p. 49, Revised; January 28, 2009 omment on Slide

4 Factoring Using 2 Method III: Using 2 Step : Simplify, Use 2 Repeat: Step until done... Unless the problem is very simple the other methods will be easier. 2nd istributive Law (2) X + cd = (X+c)(X+d) Example + X Use (2) ( + X) ( + X) Use (2) again ( + X)( + X)( + X) Get extended (2) + X = ( + X)( + X)( + X) Example + Use (2) again ( + ) ( + ) Use (2) again, twice ( + ) ( + )( + ) ( + ) dig4factoringh.fm p. 50 Revised; January 28, 2009 Slide 2 Factoring Using 2 Factoring Using 2 Factoring Using 2 The expression proven on the slide is: The extended (2) + X = ( + X)( + X)( + X) The dual is the extended () ( + + )X = X + X + X Example + (2) ( + ) ( + ) (2) (2) ( + ) ( + )( + ) ( + ) ( + )( + ) ( + ) ( + = ) If we had started by using the Swap rule, we would have a simpler answer in one step PROLEM Factor + + arleton University dig4factoringh.fm p. 5, Revised; January 28, 2009 omment on Slide 2

5 Factoring Using uality Method II: Factoring Using uality Step. Simplify and use if possible ut for F dual Step 2. Take the dual; get a factored, or semi factored form Step 3. Multiply out the dual to get sum-of products. The right one Step 4. Take the dual back to get the factored form. Factoring Using uality The expression to factor is Σ of Π F = + + () Use F = ( + ) + (2) Take its dual to get Π of Σ. F = [ ( + { })] + { } racket LL N terms F UL = [ + ( { + })] { + } (3) Multiply out to get Σ of Π. See box F UL = (4) Take the dual back F = ( + )( + )( + )( + ) Get the desired Π of Σ. + + = F = ( + )( + )( + )( + ) Multiply Out etails F UL =[ + { + }] { + } () = [ + + ] { + } = [ + + ] () +[ + + ] bdx+bd=bd F UL = Multiplying out is based on (). Easier for people, than factoring based on (2). lgebra of one is the dual of the algebra of the other dig4factoringh.fm p. 52 Revised; January 28, 2009 Slide 3 Factoring Using uality hanging Factoring into Multiplying Out hanging Factoring into Multiplying Out Factoring is onverted to Multiplying Out, its ual Problem We take a factoring problem which is confusing, because factoring is based on (2). This law is not a familiar high-school type algebraic law and is harder to work with. In the dual space, the dual expression is already factored. The problem is transformed into multiplying out, which is based on the first distributive law (). () is more familiar, and hence multiplying out is usually easier than factoring. Multiplying out in the dual space does not give the answer. One take the dual to get the answer. This will then be the factored form of the original expression PROLEM Show algebraically that F = takes only 8 letters or 2 gate inputs in factored form. arleton University dig4factoringh.fm p. 53, Revised; January 28, 2009 omment on Slide 3

6 Factoring Using uality Method II: Factoring Using uality Example: equal F = First use () F = ( + ) + ( + ) F =[ ({ } + { })] + [ ({ } + )] Take dual F dual = [ + ({ + } { + })] [+ ({ + } )] Multiply out = [ + ( + )( + )] [ + ( + )] Sw = ( + )) + ( + )( + ) Sw = ( + )) + ( + ) = First use () twice Put brakets around all the N terms Minus 25% if you don t differentiate between F and F dual Rearrange to use Swap Use Swap (+stuff)(+junk)= junk+ stuff Use Swap (+)(+)= + Use () F dual = heck on map Take dual back OK on map F = ( + + )( + + )( + + )( + + ) Factored form -> F = ( + + )( + + )( + + )( + + ) map of F dual dig4factoringh.fm p. 54 Revised; January 28, 2009 Slide 4 Factoring Using uality Method II: Factoring In the ual Space Method II: Factoring In the ual Space Example; Factor F = F = ( +) + ( + ) F = ( +) + ( + ) F = [+ ( + )] [+( + )] F = [+{ ( + )}] [+({ } + { })] Put brackets around all Ns ready to take dual Take the dual F d = [ {+( )}] + [ ({+} {+})] Remove extra brackets F d = {+( )} + {+} {+} Use () In General F d = + + {+} {+} Simplify, use F Use Swap d = + + [ + ] and maybe Swap after taking the Use () F d = dual F d = Map shows no more simplifications Take the dual back F = ( + ) ( + + )( + + )( + + ) 4-. PROLEM Factor EF + E + E + EF Use () Use () Use Swap In General Simplify, use and maybe Swap before taking the dual arleton University dig4factoringh.fm p. 55, Revised; January 28, 2009 omment on Slide 4

7 Factoring Karnaugh Maps and emorgan Method III: Factoring Using a K-Map Step. Plot function F on a K-map Step 2. Plot F by interchanging 0 on the map. Step 3. ircle the map to get F. Step 4. Write out the expression for F. Step 5. Use emorgan to get back F in factored form. Example: Given F = ( Σ of Π expression) F = () Plot it on a map. (2) Make a map for F, It has where F had 0 (3) ircle the F map (4) Write out the equation for F F = (5) Invert F using emorgan s law to get F as Π of Σ F = ( + )( + + )( + + )( + + ) dig4factoringh.fm p. 56 Revised; January 28, 2009 Slide 5 Factoring Karnaugh Maps and emorgan Method III: Factoring Using Karnaugh Method III: Factoring Using Karnaugh Maps This method is probably the easiest, and least error prone, for up to four variables. Five variables is at least twice the work of four. bove 5 it gets very messy. It is very easy to incorporate don t cares with this method PROLEM Factor EF + E + E + EF Using a Karnaugh map and compare your answer with the previous problem if you did it PROLEM Factor Use a Karnaugh map and obtain the minimum Π of Σ expression PROLEM Show, using a Karnaugh map, that F = takes only 8 letters or 2 gate inputs in factored form. ompare with Problem 4-0. arleton University dig4factoringh.fm p. 57, Revised; January 28, 2009 omment on Slide 5

8 Factoring Using K-Maps and emorgan Factoring Using Karnaugh Maps Steps: Given F = ( Σ of Π expression) F = Not the minimum but it doesn t matter () Plot it on a map. (2) Make a map for F, It has where F had 0 (3) ircle the F map (4) Write out the equation for F F = + + (5) Invert F using emorgan s law to get F as Π of Σ F = ( + )( + + )( + ) dig4factoringh.fm p. 58 Revised; January 28, 2009 Slide 6 Factoring Using K-Maps and emorgan Method III: Factoring Using Karnaugh Example: Factoring a 5-Variable Expression without using a map! Method II: uses initially, then duality and swap before the final cleanup. Take dual Multiply Out Factor F = E Use F = ( + + E) + ( +) F = [ ( +{ } +{ E})] +[ ({ } +)] F dual = [ + ( { + } { + E})] [ + ({ + } )] F dual = [ { + } ] + [ { + } { + E}] F dual = [ { + } ] + [{ + } { + E}] [{ E} + { }] Use ut brackets around LL N terms + Use Swap Ready to use Swap Use Swap F dual = [ { + } ] + [{ E} + { }] F dual = + + E + Use () twice Take dual back get (dual) 2 = F = (++)(++)(+++E)(+++) arleton University dig4factoringh.fm p. 59, Revised; January 28, 2009 omment on Slide 6

9 Warnings on t say (a+b+c)(e+ab)(a+d) is already Π of Σ lways simplify. Look for x+xz = x, x + xy = x + Y, xy + xy = Y before and after each step. Pick the best method: ) for 5 variables or under use general emorgan and a map. ) for 6 variables or more, use algebra or find a computer program. For multiplying out algebraically Take the dual. Simplify and use Take the reverse dual. Look for complemented letters, use swap, and simplify. Use for what is left. For factoring algebraically EFORE you take the dual Simplify and use Take the dual use Look possible Swap, simplify Use and simplify Take the reverse dual Taking the dual The hard part is getting the brackets around the LL the N terms. Try: (+ + E ) + ( +) dig4factoringh.fm p. 60 Revised; January 28, 2009 Slide 7 Original the slide above said, on t say (a+b+c)ab(a+d) is already Π of Σ." ctually it was Π of Σ, and the slide embarrased the lecturer PROLEM Explain why the (a+b+c)ab(a+d) is Π of Σ. Hint: take the dual. Method III: Factoring Using Karnaugh 4-6. PROLEM ON ULS Take the dual of F = ( + + E ) + ( + ) To check your answer look in the omment on Slide 6 I ll bet you can t get it right the first time without looking. arleton University dig4factoringh.fm p. 6, Revised; January 28, 2009 omment on Slide 7

10 dig4factoringh.fm p. 62 Revised; January 28, 2009 Slide 8 Method III: Factoring Using Karnaugh arleton University dig4factoringh.fm p. 63, Revised; January 28, 2009 omment on Slide 8