1

enter image description here

Hi, I am junior in college and having trouble with my computer architecture classwork. Anyone care to help & tell me if I got them right?

Question1. Convert truth table into bool equation.

Question2. Find miminum SOP(sum of products)

Question3. Use K-map(Karnaugh map) to simplify.

Kit Ostrihon
  • 824
  • 2
  • 14
  • 36
Bossam
  • 744
  • 2
  • 9
  • 24
  • In 2), `F(X,Y,Z) = XY` cannot be right: The second line in the truth table already does not match. – JimmyB Mar 16 '16 at 16:18
  • `(!x*z + x*!z)` is *not* `0`. Also, `(x + !x) = 1`, but `(...)+1 = 1`. – JimmyB Mar 16 '16 at 16:30
  • Your result for 3) also cannot be correct: Line #6 in the table says that `F(1,0,1) = 0`, while by your answer `F(1,0,1) = 1`. – JimmyB Mar 16 '16 at 16:45
  • How did you get to `x * y * !z` from the K-map? – JimmyB Mar 16 '16 at 16:52
  • Is the third question even relevant to the given truth table or is the function on the bottom right a different example of an expression to be simplified by using a K-map? – Kit Ostrihon Mar 17 '16 at 16:26

1 Answers1

1

You can simplify the original expression matching the given truth-table just by using Karnaugh maps:

K-map of the original expression simplified with minimal DNF and minimal CNF marked out - generated using latex

f(x,y,z) = ∑(1,3,4,6,7) = m1 + m3 + m4 + m6 + m7
         = ¬x·¬y·z + ¬x·y·z + x·y·z + x·¬y·¬z + x·y·¬z      //sum of minterms

f(x,y,z) = ∏(0,2,5) = M0 · M2 · M5
         = (x + y + z)·(x + ¬y + z)·(¬x + y + ¬z)           //product of maxterms

f(x,y,z) = x·y + ¬x·z + x·¬z                               //minimal DNF
         = (x + z)·(¬x + y + ¬z)                           //minimal CNF

You would get the same result using the laws of Boolean algebra:

¬x·¬y·z  + ¬x·y·z + x·y·z  + x·y·¬z + x·¬y·¬z
¬x·(¬y·z +   y·z) + x·(y·z + y·¬z   + ¬y·¬z)      //distributivity
¬x·(z·(¬y +   y)) + x·(y·(z + ¬z)   + ¬y·¬z))     //distributivity
¬x·(z·(    1   )) + x·(y·(  1   )   + ¬y·¬z))     //complementation
¬x·(z           ) + x·(y            + ¬y·¬z))     //identity for ·
¬x·(z           ) + x·(y +  y·¬z    + ¬y·¬z))     //absorption
¬x·(z           ) + x·(y +  ¬z·(y   +    ¬y))     //distributivity
¬x·(z           ) + x·(y +  ¬z·(    1      ))     //complementation
¬x·(z           ) + x·(y +  ¬z)                   //identity for ·
¬x·z              + x·y  +  x·¬z                  //distributivity

¬x·z + x·y + x·¬z                                 //minimal DNF

¬x·z + x·y + x·¬z
¬x·z + x·(y + ¬z)                                 //distributivity
(¬x + x)·(¬x + (y + ¬z))·(z + x)·(z + (y + ¬z))   //distributivity
(   1  )·(¬x +  y + ¬z )·(z + x)·(z +  y + ¬z)    //complementation
(   1  )·(¬x +  y + ¬z )·(z + x)·(y + 1)          //complementation
(   1  )·(¬x +  y + ¬z )·(z + x)·(1)              //annihilator for +
         (¬x +  y + ¬z )·(z + x)                  //identity for ·

         (¬x + y + ¬z)·(x + z)                    //minimal CNF
Kit Ostrihon
  • 824
  • 2
  • 14
  • 36
  • Do you have a tool to draw such nice Karnaugh maps? – Axel Kemper Mar 30 '16 at 12:25
  • 1
    Sorry, I mentioned it only in the picture's description - it was generated using latex and a [\karnaughmap command](http://mirrors.nic.cz/tex-archive/macros/latex/contrib/karnaugh/kvdoc.pdf). – Kit Ostrihon Mar 30 '16 at 12:52
  • Meanwhile, I have found a nice [online tool](http://www.mathematik.uni-marburg.de/~thormae/lectures/ti1/code/karnaughmap/) to experiment with Karnaugh maps. – Axel Kemper Aug 26 '16 at 07:55