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Binary Numbers 1s_complement 2s_complement Binary Subtraction Binary Sub. Ex's Sign_magnitude SignM EX Gray Coding BCD coding Digital gates NAND NOR & XNOR Theorems Boolean Functions BFunc Examples Minterm Maxterm Sum of Minterms Prdt of Maxterms 2 var K-map 3 var K-map 4 var K-map 5 var K-map Prime Implicant PI example K-map Ex's KMap minimization 2 var EX

3 variable K-map, Example -3

Minimize following

 

F(x,y,z) =     (3,4,6,7)

 

Above is a common format of representing the K-map problems. The numbers 3,4,6,7 are the location of cells in the 3-var k-map table. Discussed below is a 3 var K– map with 1 and 0 values assigned to cells.

00

01

4

5

11

10

7

6

 

x’y’z’ = 0

 

 

x’y’z = 0

 

 

x’yz = 1

 

 

x’yz’ = 0

 

 

xy’z’ = 1

 

 

xy’z = 0

 

 

xyz = 1

 

 

xyz’ = 1

 

0

1

x

yz

Interview Questions. Main, FPGA, Digital Fundamentals
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.    

Any adjacent 1, 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’ = 0

 

 

x’y’z = 0

 

 

x’yz = 1

     

 

x’yz’ = 0

 

 

xy’z’ = 1

 

 

xy’z = 0

 

 

xyz = 1

 

 

xyz’ = 1

 

0

1

x

yz

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.
F = (x’yz+ xyz) + (xyz’ + xy’z’)
F = (yz + xz’)  ... Final Answer.
x’y’z’
x’y’z
x’yz
x’yz’
xy’z’
xy’z
xyz
xyz’
Solved Examples
1. F(x,y,z) =     (0,1,6,7) - Minimization, on this page.
2. F(x,y,z) =     (0,1,4,5,6,7) - Minimization from here.
3. F(x,y,z) =     (3,4,6,7) - Minimization from here.
4. F(x,y,z) =     (0,1,2,3,4,5,6,7) - Minimization from here.
MINIMIZATION USING THREE VARIABLE KARNAUGH MAP