BOOLENA ALGEBRA AND LOGIC GATES
Step 3: Step 3 is shown in Table 4.62.3.
Table E4.62.3 Combination of four minterms Minterm Group Binary A B C D 1, 3, 9, 11 – 0 – 1 PI 1, 9, 3, 11 – 0 – 1 PI Eliminated 2, 3, 10, 11 – 0 1 – PI 2, 10, 3, 11 – 0 1 – PI Eliminated 3, 7, 11, 15 – – 1 1 PI 3, 11, 7, 15 – – 1 1 PI Eliminated 9, 11, 13, 15 1 – – 1 PI 9, 13, 11, 15 1 – – 1 PI Eliminated 12, 13, 14, 15 1 1 – – PI 12, 14, 13, 15 1 1 – – PI Eliminated STEP 2:
Compare every term of the lowest group with each term in the adjacent group. If two minterms differ in only one variable, that variable should be removed and a dash (–) is placed at the position, thus a new term with one less literal is formed. If such a situation occurs, a check mark () is placed next to both minterms. After all pairs of terms have been considered, a horizontal line is drawn under the last terms as depicted in Table E4.62.2.
STEP 3:
Now we repeat step-2 with newly formed groups i.e. we combine four minterms of adjacent groups if possibilities exist. In this case, dashes (–) exist in same position of two groups and only one position will be different. Table E4.62.3 shows the combination of four minterms.
Chapter 4 Minimization Techniques Page 287
REVIEW QUESTIONS
1. What are SOP and POS forms ? Explain with an example. 2. What are min terms and max terms. In which form of
expressions do they occur ? Give one example of expression in the form of min terms and max terms.
3. How do you convert a standard POS form into a standard SOP form and vice-versa ?
4. Name the two basic types of boolean expressions and explain each with an example.
5. Define Maxterm and Minterm.
6. What is the use of Karnaugh map ? How is it drawn ? Give example. What is the difference between this map for min terms and max terms ?
7. Explain the procedure for grouping of cells in Karnaugh map.
8. What is meant by Don’t care conditions ? Give example. 9. Explain the K-map reduction technique.
10. What is variable mapping technique ? What is its advantage ?
11. Discuss the features of the Quine-McCluskey method. 12. State the advantage of Quine-McCluskey technique over
K-map technique.
13. Explain complete steps used in Quine-McClusky minimization techniques for simplifying the given expression. 14. Write short note on incompletely specified functions.
REVIEW PROBLEMS
15. Convert into other canonical form (POS).
, , , ,
y=SM 0 5 1 8 9_ i
16. Write minterm and maxterm Boolean functions expressed by f A B C_ , , i=P0,3,7.
17. Find the minterm and canonical sum of products (SOP) form of the switching function f A B C_ , , i whose truth table is given as follows : A B C f A B C_ , , i 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1
18. Find the maxterms and canonical product of sum (POS)
form of the switching function f A B C_ , , i whose truth table is given in Table 2.20.
19. Find the sum of minterms and product of maxterms expression for the switching function f A B C_ , , i=A+BC. 20. Convert the following to the other canonical form.
(a) F x y z_ , , i=S_1 3 7, , i
(b) F A B C D_ , , , i=S_0 2 6 11 13 14, , , , , i
(c) F x y z_ , , i=P_0 3 6 7, , , i
(d) F A B C D_ , , , i=P_0 1 2 3 4 6 12, , , , , , i
21. Given F A B C D_ , , , i=Sm 0 1 2 6 7 13 15_ , , , , , , i.
(a) Find the minterm expansion for F (both decimal and algebraic form).
(b) Find the maxterm expansion for F (both decimal and algebraic form).
22. Express each of the following functions by a minterm canonical formula without first constructing a truth table.
(a) f x y z_ , , i=x y_ +zi+z (b) f x y z_ , , i=_x+y xi_ +zi
23. Express each of the following functions by a maxterm canonical formula without first constructing a truth table.
(a) f x y z_ , , i=_y+z xyi_ +zi (b) f x y z_ , , i= +x x z y_ +zi
24. Transform each of the following canonical expressions into its other canonical form in decimal notation.
(a) f x y z_ , , i=Sm 1 3 5_ , , i
(b) f x y z_ , , i=PM 3 4_ , i
(c) f w x y z_ , , , i=Sm 0 1 2 3 7 9 11 12 15_ , , , , , , , , i
(d) f w x y z_ , , , i=PM 0 2 5 6 7 8 9 11 12_ , , , , , , , , i
25. Plot on Karnaugh map
, , , , , , , , , ,
Y A B C D_ i=Sm 0 2 3 6 8 9 14 15_ i
26. Plot on K-map f A B C D_ , , , i=PM 0 4 6 8 10 12 14_ , , , , , , i
27. The sum of all minterms of a Boolean function of n variables to 0.
(a) Prove the above statement for n=3.
28. The product of all maxterms of a Boolean function of n variables is 0.
(a) Prove the above statement for n=3.
29. Represent each of the following Boolean functions on a Karnaugh map
(c) f w x y z_ , , , i=Sm 1 6 7 8 10 12 14_ , , , , , , i
(d) f w x y z_ , , , i=PM 0 3 4 7 9 13 14_ , , , , , , i
(e) f x y z_ , , i=xy+xy+yz (f) f x y z_ , , i=_x+z yi_ +z yi_ +zi
30. Obtain the simplified SOP and POS expression for the following using K-map.
(i) Y A B C D_ , , , i=S_0 2 3 5 6 8 10 13 14, , , , , , , , i
(ii) Y A B C D_ , , , i=S_1 4 7 9 11 12 15, , , , , , i
(vi) Y A B C_ , , i=P_5 7, i
Page 288 Minimization Techniques Chapter 4
Digital Electronics by Ashish Murolia and RK Kanodia For More Details visit www.nodia.co.in following using K-map.
(i) Y A B C D_ , , , i=S_0 3 5 8 9 13, , , , , i+d_1 4 7 12, , , i
(ii) Y A B C D_ , , , i=S_0 3 4 6 8 9 14, , , , , , i+d_1 2 5 15, , , i
(iii) Y A B C D_ , , , i=P_0 3 5 8 9 13, , , , , i+d_1 4 7 12, , , i
(iv) Y A B C D_ , , , i=P_0 3 4 6 8 9 14, , , , , , i+d_1 2 5 15, , , i
32. Obtain the simplified expressions in sum of products for the following Boolean functions :
(a) xy+x y z++xyz
(b) AB+BC+B C (c) a b+bc+abc
(d) xyz+xyz+xyz+xyz
33. Obtain the simplified expressions in sum of products for the following Boolean functions :
(a) D A_ +Bi+B C_ +ADi
(b) ABD+A C D+AB+ACD+AB D (d) A B C D+AC D+BCD+ABCD+BC D (e) xz+wxy+w xy_ +xyi
34. Obtain the simplified expressions in (1) sum of products and (2) product of sums :
(a) x z+y z+yz+xyz
(c) _A+ +B D Ai_ + +B C Ai_ + +B D Bi_ + +C Di (d) _A+ +B D Ai_ +D Ai_ + +B D Ai_ + + +B C Di 35. Simplify the Boolean function F in sum of products using
the don’t-care conditions d : (a) F= +y x z, d=yz+xy
(b) F=B C D+BCD+ABCD, d=BCD+ABC D 36. Simplify the Boolean function F using the don’t care
conditions d, in (1) sum of products and (2) product of sums:
(a) F=ABD+ACD+ABC d=ABC D+ACD+AB D (b) F=w xy_ +x y+xyzi+x z y_ +wi
d=wx yz_ +yzi+wyz
37. Using Karnaugh maps, determine all the minimal sums and minimal products for each of the following Boolean functions.
(a) f w x y z_ , , , i=Sm 0 1 6 7 8 14 15_ , , , , , , i
(c) f w x y z_ , , , i=Sm 1 3 4 5 10 11 12 14_ , , , , , , , i
(f) f w x y z_ , , , i=PM 4 6 7 8 12 14_ , , , , , i
(g) f w x y z_ , , , i=w xz+xyz+wxz+xy z
38. Using Karnaugh maps, determine all the minimal sums and minimal products for each of the following Boolean functions.
(a) f w x y z_ , , , i=Sm 0 2 6 7 9 10 15_ , , , , , , i
(d) f w x y z_ , , , i=PM 0 2 6 8 10 12 14 15_ , , , , , , , i
(g) f w x y z_ , , , i=wx+yz+wx y+wxyz
(i) f w x y z_ , , , i=_w+x wi_ + +y z wi_ + +x z wi_ + +y zi
39. Reduce the following expressions using K-map and
implement them in universal logic. (a) Sm 5 6 7 9 10 11 13 14 15_ , , , , , , , , i
(c) PM_1 4 5 11 12 14, , , , , id_6 7 15, , i
40. Reduce the following expressions using K-map and implement them in universal logic.
(d) PM_3 6 8 11 13 14, , , , , id_1 5 7 10, , , i
(e) Sm_0 1 4 5 6 7 9 11 15, , , , , , , , i+d_10 14, i
41. Minimize the following expression using K-map and realize using NOR gates only :
, , , , , , , , , ,
f P Q R S_ i=PM 1 4 6 9 10 11 14 15_ i
42. Obtain the minimal SOP expression for
, , , , , , , , ,
m 2 3 5 7 9 11 12 13 14 15
S _ i and implement it in NAND
logic.
43. Obtain the minimal POS expression for
, , , , , , , , , , ,
M 0 1 2 4 5 6 9 11 12 13 14 15
P _ i and implement it in
NOR logic.
44. Using K-map, simplify the equation :
, , , , , , , , , , ,
F A B C D_ i=Sm_0 2 8 9i+d_3 7 10 11 14 15i
Obtain the minimal SOP expression and implement it in NAND logic.
45. Using a Karnaugh map, determine a minimal sum and a minimal product for each of the following functions.
(a) f v w x y z_ , , , , i =Sm 1 5 9 11 13 20 21 26 27 28 29 30 31_ , , , , , , , , , , , , i (b) f v w x y z_ , , , , i , , , , , , , , , , , , , M 0 2 4 6 8 12 14 15 16 18 20 22 30 31 P = _ i
46. For each of the following Boolean functions, determine a minimal sum and a minimal product using variable entered map technique.
(a) ABC+ABCD+ABC D+ABC
(c) A BCD+A BCD+ABCD+ABCD+ABCD
+ABC D+ABC D+ABC D 47. Simplify the following using the tabular method.
(i) Y A B C D_ , , , i=S_0 2 4 5 7 8 10 12, , , , , , , i
(ii) Y A B C D_ , , , i=S_0 1 2 3 4 6 8 10 12 14, , , , , , , , , i
48. Reduce the following using Quine-McCluskey method: (a) S0,1,2,4,5,6,8,9,14
(b) S1,2,8,9,10,14+d 0 3 6 11_ , , , i
(d) P_0 1 6 7 9 10 14, , , , , , i+d_2 3 8, , i
49. Using the Quine-McCluskey method, obtain all the prime implicants for each of the following Boolean functions. (d) f w x y z_ , , , i=Sm_0 1 2 6 7 9 10 12, , , , , , , i+dc_3 5, i
(b) f w x y z_ , , , i=PM_0 2 3 4 5 12 13, , , , , , i+dc_8 10, i
50. For each function, find a minimum sum-of-products solution using the Quine-McCluskey method.
(b) f a b c d_ , , , i=Sm_0 1 5 6 8 9 11 13, , , , , , , i+Sd_7 10 12, , i
(c) f a b c d_ , , , i=Sm_3 4 6 7 8 9 11 13 14, , , , , , , , i+Sd_2 5 15, , i