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Which of the following functions implements the Karnaugh map shown below? ‘X’ being don’t care condition. CD AB 00 01 11 10 00 0 0 1 0 01 X X 1 X 11 0 1 1 0 10 0 1 1 0

Which of the following functions implements the Karnaugh map shown below? ‘X’ being don’t care condition. CD AB 00 01 11 10 00 0 0 1 0 01 X X 1 X 11 0 1 1 0 10 0 1 1 0 Correct Answer D ( C + A)

Concept:

Karnaugh maps

We can simplify the Boolean expressions with the help of the Boolean laws but when the variables increase the minimization takes much time so the new technique is introduced to reduce the time.

This method is known as the Karnaugh map method or K-map method. It is a graphical method, which consists of 2n cells for ‘n’ variables. The adjacent cells have differed only in single bit position.

The structure of the K – maps for the different number of variables are shown below:

2 variable K - map

Possibility of grouping 4 minterms or 2.

3 variable K – map

Possibility of grouping min - terms like 8, 4, 2.

  • The possible combinations of grouping 4 adjacent min terms are {(m0, m1, m3, m2), (m4, m5, m7, m6), (m0, m1, m4, m5), (m1, m3, m5, m7), (m3, m2, m7, m6) and (m2, m0, m6, m4)}.
  • The possible combinations of grouping 2 adjacent min terms are {(m0, m1), (m1, m3), (m3, m2), (m2, m0), (m4, m5), (m5, m7), (m7, m6), (m6, m4), (m0, m4), (m1, m5), (m3, m7) and (m2, m6)}.
  • If x=0, then 3 variable K-map becomes 2 variable K-map.

 

4 variable  K – map

Possibility of grouping 16, 8, 4, and 2.

Calculation:

In the given K – map considering the A as MSB and D as the LSB the following two groupings can be done as shown.

FIG

The first grouping gives the expression CD and the second grouping gives the expression AD.

The final expression is sum of two that is:

F = CD + AD

F = D(C + A)

Option 2 is correct.

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