Don’t care conditions in Karnaugh Map (K map):

 

Don’t care conditions in Karnaugh Map (K map):

What is don’t care condition?

We have discussed about solving and minimizing SOP and POS expressions using K map. The output is specified as ‘1’ or ‘0’. There are some conditions where output cannot be specified, so we use don’t care conditions in that case. It is represented as ‘X’.

For example in Binary Coded Decimal (BCD) the output is specified from 0 to 9. The output for the inputs above 9 is not specified. So in this case we use don’t care conditions. 

In Excess 3 code, the output is not specified for 0000, 0001, 0011. In that case we use don’t care conditions.

Rules for using don’t care conditions in K map:

1. First fill the cells with 1’s in SOP and 0’s in POS. Then mark the cells with ’X’ for don’t care cells specified in the function.

2. While grouping in SOP don’t care ‘X’ is considered as ‘1’ and in POS ‘X’ is considered as ‘0’.

3. If the adjacent cell consists of ‘X’, it can be grouped along.

4. Write the Minterm and Maxterm for each group. 

5. Thus the minimized expression is formed.

Example 1:

Minimize the SOP expression using K map(with don’t care conditions):

F(A,B,C,D)=∑m(1,5,6,12,13,15)+ d(2,4)

K map with don't care condition

• First ‘1’ is filled in the cells 1,5,6,12,13 and 14. The cells 2 and 4 are filled with ‘X’.
• The cells 2 and 4 are grouped along with the adjacent cells.
• There are 4 groups.

1. In group 1 cells 1 and 5 are grouped and the common Minterm is A'C'D.

2. In group 2 cells 2 and 6 are grouped and the common Minterm is A'CD'.

3. In group 3 cells 13 and 15 are grouped and the common Minterm is ABD.

4. In group 4 cells 4,5,12 and 13 are grouped and the common Minterm is BC'.

So the minimized expression from K map is

F(A,B,C,D)= A'C'D+ A'CD'+ ABD+ BC'

Example 2:

Minimize the POS expression using K map(with don’t care conditions):

f(A,B,C,D)=πM(1,5,6,12,13,15)+ d(2,4)

K map with don't care condition for POS expression

• First ‘0’ is filled in the cells 1,5,6,12,13 and 14. The cells 2 and 4 are filled with ‘X’.
• The cells 2 and 4 are grouped along with the adjacent cells.
• There are 4 groups.

1. In group 1 cells 1 and 5 are grouped and the common Maxterm is A+C+D'.

2. In group 2 cells 2 and 6 are grouped and the common Maxterm is A+C'+D.

3. In group 3 cells 13 and 15 are grouped and the common Maxterm is A'+B'+D'.

4. In group 4 cells 4,5,12 and 13 are grouped and the common Maxterm is B'+C.

So the minimized expression from K map is

f(A,B,C,D)= (A+C+D').(A+C'+D).(A'+B'+D').(B'+C)