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3 variable K-map - Examples

Minimize following function.

 

F(x,y,z) =     (0,1,6,7)

 

Above is a common format of representing the K-map problems. The numbers 0,1,6,7 are the location of cells in the 3-var k-map table discussed below.  3 variable K– map with 1 and 0 values assigned to cells is shown in table below.

 

00

01

4

5

11

10

7

6

 

x’y’z’ = 1

 

 

x’y’z = 1

 

 

x’yz = 0

 

 

x’yz’ = 0

 

 

xy’z’ = 0

 

 

xy’z = 0

 

 

Xyz = 1

 

 

xyz’ = 1

 

0

1

x

yz

Any adjacent 2, 4 or 8 cells can be grouped to find a minimized logic value. Following plot will show grouping of adjacent cells.

00

01

4

5

11

10

7

6

 

x’y’z’ = 1

 

 

x’y’z = 1

 

 

x’yz = 0

     

 

x’yz’ = 0

 

 

xy’z’ = 0

 

 

xy’z = 0

 

 

Xyz = 1

 

 

xyz’ = 1

 

0

1

x

yz

F = x’y’(z+z’) + xy(z+z’)
F= x’y’ + xy .. Final Answer.
1. F(x,y,z) =     (0,1,6,7) - Minimization, discussed on this page.
2. F(x,y,z) =     (0,1,4,5,6,7) - Access minimization from here.
3. F(x,y,z) =     (3,4,6,7) - Access minimization from here.
4. F(x,y,z) =     (0,1,2,3,4,5,6,7) - Access minimization from here.
Following list of functions are minimized using 3 var K-map’s.
The K-map for 3 variables is plotted above. You will notice the column for 11 and 10 is inter-changed. This is done to allow only one variable to change across adjacent cells. This adjustment in columns allows in minimization of logic mapped into tables.    
The two step minimization equation is shown below.
With reference to the table above the cells under the dotted box’s can be combined to come-up with following reduced  equation.
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