Chapter 2 : Part 2

Q 23. Simplify the function using Karnaugh map and implement using minimum Lumber of logic gates.

                           F =  (2, 9, 10, 12, 13) + D(1, 5, 14)

What are the limitations of Karnaugh map?

Ans.                  F =  (2, 9, 10, 12, 13) + D(1, 5, 14)

K-map:

                          

Implementation using minimum number of logic gates can be obtained from the minimized output of the given function.

Implementation is as shown:

   

            

Limitations of K-map:

For large number of variables i e more than six variables the K-map becomes cumbersome It is difficult to solve the output of K-map having 7 8 and more variables as it covers more space and need large time for calculations Also, the K-map for 6 variables is possible and for more variables Q-M method or tabular minimization method is used

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Q 24 Minimize the following four variable functions using sum-of-products

Karnaugh Maps

f(a, b, c, d) = f1(a, b, c, d)   f2(a, b, c, d)

where f1 = a’ d + bc + b’ c’ d’ +  (1, 2, 11, 13, 14, 15)

and f2 =  (0, 2, 4, 8, 9, 10, 14) d (1, 7, 13, 15)

Ans.

           

        

                                 

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Q 25. Minimize the following function by Quine Mccluskey method and

list all prime implicants of essential prime implicants. Is the minimum SOP unique, if not all the minimal solutions for the functions?

F (a,b,c,d,e,f) =  (0,2,4,7,8,16,24,32,36,40,48) + d (5,18,22,23,54,56)

Ans. Arrange the minterms according to no. of l’s

Group the minterms into group of two

Group of minterms into group of four.

Group of minterms in Group of eight

Table of prime impilcants:

Thus, output

                      

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Q 26. Design a combinational circuit which has four inputs and one output. The output is equal to 1 when

(i) All the inputs equal to 1 or

(ii) None of the inputs equal to 1.

(iii) An odd number of inputs are equal to 1.

Draw the logic circuit using minimum number of NAND gates.

Ans.

Truth Table

                                    

Implementation using NAND gates:

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Q 27. Obtain the set of prime implicants for  (0,1,3,4,6,7,8,9, 14,15) using the binary designations of mm-terms using Q — M method.

Ans. Using Q-M method  (0,1,3,4,6,7,8,9, 14,15)

Step 1

                               

Step 2

                             

Step 3

                 

Step 4

                      

Step 5

                             

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Q 28. Obtain the set of prime implicants for  (0, 1, 6, 7, 8, 9, 13, 14, 15) using the binary designations of minterms using Q-M method.

Ans.

Step 1:

                

Step 2:

                    

Step 3:

                  

Step 4:

                  

Step 5: