1. ECB2212-Digital Electronics
Ms.K.Indra Gandhi
Asst Prof (Sr.Gr) /ECE

2. Examples of Solved Problems
CprE 281: Digital Logic
Iowa State University, Ames, IA
Copyright © Alexander Stoytchev

3. Determine if the following equation is valid
Example 1
Determine if the following equation is valid

4. ?

5. ?
LHS
RHS

6. Left-Hand Side (LHS)

7. Left-Hand Side (LHS)

8. Left-Hand Side (LHS)

9. Right-Hand Side (RHS)

10. Right-Hand Side (RHS)

11. Right-Hand Side (RHS)

12. ?
LHS
RHS

13. Design the minimum-cost product-of-sums expression for the function
Example 2
Design the minimum-cost product-of-sums expression for the function
f(x1, x2, x3) = Σ m(0, 2, 4, 5, 6, 7)

14. Minterms and Maxterms (with three variables)
[ Figure 2.22 from the textbook ]

15. Minterms and Maxterms (with three variables)
The function is
1 for these rows

16. Minterms and Maxterms (with three variables)
The function is
1 for these rows
The function is
0 for these rows

17. Two different ways to specify the same function f of three variables
f(x1, x2, x3) = Σ m(0, 2, 4, 5, 6, 7)
f(x1, x2, x3) = Π M(1, 3)

18. The POS Expression f(x1, x2, x3) = Π M(1, 3) = M1 M3
= ( x1 + x2 + x3)( x1 + x2 + x3)

19. The Minimum POS Expression
f(x1, x2, x3)
= ( x1 + x2 + x3)( x1 + x2 + x3)
= ( x1 + x3 + x2)( x1 + x3 + x2)
= ( x1 + x3 )
Hint: Use the following Boolean Algebra theorem

20. Alternative Solution Using K-Maps

21. Alternative Solution Using K-Maps

22. Alternative Solution Using K-Maps

23. Alternative Solution Using K-Maps

24. Alternative Solution Using K-Maps1

25. Alternative Solution Using K-Maps1
( x1 + x3 )

26. Example 3

27. Condition A

28. Condition A

29. Condition B

30. Condition B

31. Condition C

32. Condition C

33. The output of the circuit can be expressed as f = AB + AC + BC

34. The output of the circuit can be expressed as f = AB + AC + BC

35. The output of the circuit can be expressed as f = AB + AC + BC

36. Finally, we get

37. Solve the previous problem using Venn diagrams.
Example 4
Solve the previous problem using Venn diagrams.

38. Venn Diagrams (find the areas that are shaded at least two times)x1 x2 x1 x2 x3 x3
(a) Function A
(b) Function B x1 x2 x1 x2 x3 x3
(c) Function C
(d) Function f
[ Figure 2.66 from the textbook ]

39. Design the minimum-cost SOP and POS expression for the function
Example 5
Design the minimum-cost SOP and POS expression for the function
f(x1, x2, x3, x4) = Σ m(4, 6, 8, 10, 11, 12, 15) + D(3, 5, 7, 9)

40. Let’s Use a K-Map
f(x1, x2, x3, x4) = Σ m(4, 6, 8, 10, 11, 12, 15) + D(3, 5, 7, 9)

41. Let’s Use a K-Map
f(x1, x2, x3, x4) = Σ m(4, 6, 8, 10, 11, 12, 15) + D(3, 5, 7, 9) 1 d

42. The SOP Expression
[ Figure 2.67a from the textbook ]

43. What about the POS Expression?
f(x1, x2, x3, x4) = Σ m(4, 6, 8, 10, 11, 12, 15) + D(3, 5, 7, 9) 1 d

44. The POS Expression
[ Figure 2.67b from the textbook ]

45. Example 6
Use K-maps to find the minimum-cost SOP and POS expression for the function

46. Let’s map the expression to the K-Map

47. Let’s map the expression to the K-Mapd d d

48. Let’s map the expression to the K-Mapd d d

49. The SOP Expression
[ Figure 2.68a from the textbook ]

50. What about the POS Expression?1 d

51. The POS Expression
[ Figure 2.68b from the textbook ]

52. Derive the minimum-cost SOP expression for
Example 7
Derive the minimum-cost SOP expression for

53. First, expand the expression using property 12a

54. Construct the K-Map for this expression
s1 s2 s3
s1 s2 s3

55. Construct the K-Map for this expression
[ Figure 2.69 from the textbook ]

56. Write the Verilog code for the following circuit …
Example 8
Write the Verilog code for the following circuit …

57. Logic Circuit
[ Figure 2.70 from the textbook ]

58. Circuit for 2-1 Multiplexers
(c) Graphical symbol
(b) Circuit
f (s, x1, x2) =
s x1
s x2 +
[ Figure 2.33b-c from the textbook ]

59. Logic Circuit vs Verilog Code
[ Figure 2.70 from the textbook ]
[ Figure 2.71 from the textbook ]

60. Write the Verilog code for the following circuit …
Example 9
Write the Verilog code for the following circuit …

61. The Logic Circuit for this Example
[ Figure 2.72 from the textbook ]

62. Circuit for 2-1 Multiplexers
(c) Graphical symbol
(b) Circuit
f (s, x1, x2) =
s x1
s x2 +
[ Figure 2.33b-c from the textbook ]

63. Addition of Binary Numbers

64. Logic Circuit vs Verilog Code
[ Figure 2.73 from the textbook ]

65. Questions?

66. THE END