Lecture 2 Karnaugh Maps by htt39969

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									Lecture 2: Karnaugh Maps

      Soon Tee Teoh
         CS 147
               Karnaugh Maps
• K-Maps are used to simplify Boolean
  expressions
• From truth table to K-map:


                                    y
X    Y     F                                0       1
                            x
0    0     1
                                0
                                        1       0
0    1     0
1    0     1                    1       1       0
1    1     0
              Simplifying Expression
• Find blocks (rectangles) of 1s in K-map



                              This column represents Y=0

          y                   F=   x’y’ + xy’     =   y’
                  0       1
  x

      0
              1       0
                               From truth table   From K-map
      1       1       0
                3-Variable K-map
• K-map with 3 variables
                      yz
                  x         00      01      11       10

                      0

                      1




    Exercise: (1) Draw the block for x’y, (2) Draw the block y,
              (3) Draw the block z’

   Exercise: Draw the K-map for the expression f = x’yz’ + xy’z’ +xy’z
        3-Variable K-map

x   y   z   f
0   0   0   1
                    yz
0   0   1   1   x            00     01    11   10

0   1   0   0       0    1         1      1    0
0   1   1   1       1    1         1      1    0
1   0   0   1
1   0   1   1
                             F = Y’ + Z
1   1   0   0
1   1   1   1
     4-variable K-map
     yz
              00       01       11       10
wx

          0        1        0        0
00


01
          1        1        0        1

11        1        1        0        1

10        0        1        0        0
     4-variable K-map
     yz
              00       01       11       10
wx

          1        1        0        0
00


01
          1        1        0        0

11        1        1        0        0

10        1        1        0        0
                    Implicants
• Any single or group of 1s that can be
  combined is called an implicant.
• An implicant that cannot be combined with
  another implicant to eliminate a variable is
  called a prime implicant.
• A prime implicant which contains one or
  more 1s that are not contained in another
  prime implicant is called an essential
  prime implicant.
          See text (pg. 59) for more formal definitions.
                   Example
     yz
              00        01       11       10
wx

          0         1        1        0
00


01
          1         1        1        0

11        1         0        1        1

10        0         0        1        1
                   Example
     yz
              00        01       11       10
wx

          0         1        1        0
00


01
          1         1        1        0

11        1         0        1        1

10        0         0        1        1
                   Example
     yz
              00          01          11            10
wx

          0           1           1             0
00


01
          1           1           1             0

11        1           0           1             1

10        0           0           1             1

                   Is this a prime implicant?
                   Example
     yz
              00        01         11          10
wx

          0         1          1           0
00


01
          1         1          1           0

11        1         0          1           1

10        0         0          1           1

          There is a prime implicant missing: wxz’
                   Example
     yz
              00        01         11         10
wx

          0         1          1          0
00


01
          1         1          1          0

11        1         0          1          1

10        0         0          1          1

          These two prime implicants are essential
                 Strategy
• Include all essential prime implicants.
• Then add other prime implicants until all
  1s are covered.
           Don’t Care States

                         yz
                                  00       01       11       10
                    wx
• For particular
                              0        0        x        0
  input, we don’t   00
  care if output
  is 1 or 0         01
                              1        1        0        0
• Can be used
  to your           11        x        1        1        1
  advantage
                    10        0        0        1        1
               Don’t Care States

                        yz
                                 00       01       11       10
                   wx

                             0        0        x        0
                    00


F = XY’ + WY        01
                             1        1        0        0

                    11       x        1        1        1

                    10       0        0        1        1

								
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