1. Gate-Level Minimization
Chapter 3

2. The Map Method
Two-variable map

3. Two-variable map

4. Three-variable map Adjacent Adjacent
Adjacent when minterms differ by one variable

5. Graphical view of adjacency
Taking the difference between adjacent squares gives:

6. Example 3.1
Simplify the Boolean function

7. Example 3.2
Simplify the Boolean Function

8. Three variable map 00 01 11 10 1

9. Three-variable map
The number of adjacent squares that may be combined must always represent a number that is a power of two:
One square represents one minterm which in this case gives a term with three literals
Two adjacent squares represent a term with two literals
Four adjacent squares represent a term with one literal
Eight adjacent squares encompass the entire map and produce a function equal to 1

10. Three-variable map 00 01 11 10 1

11. Three-variable map 00 01 11 10 1

12. Three-variable map 00 01 11 10 1

13. Example 3.3

14. Example 3.4 Let the Boolean Function
(a) Express this function as a sum of minterms

15. Example 3.4 (b) Find the minimal sum-of-products expression 00 01 1110 1
Find adjacent squares

16. Problem 3.3 (a)
Simplify the following Boolean function, using three-variable maps 00 01 11 10 1
Find adjacent squares

17. Problem 3.3 (a) Voila! Another solution (without a three-variable map)
Factorize the expression
Voila!

18. Four-Variable Map

19. Four-variable map
The number of adjacent squares that may be combined must always represent a number that is a power of two:
One square represents one minterm which in this case gives a term with four literals
Two adjacent squares represent a term with three literals
Four adjacent squares represent a term with two literals
Eight adjacent squares represent a term with one literal
Sixteen adjacent squares encompass the entire map and produce a function equal to 1

20. Adjacency in a four-variable map

21. Adjacency in a four-variable map

22. Example 3.5 Simplify the Boolean Function 00 01 11 10 1 1 1 1 1 1 1 1

23. Example 3.6
Simplify the Boolean Function 00 01 11 10 1 1 1 1 1 1 1

24. Prime implicants
When choosing adjacent squares in a map, make sure that:
All the minterms are covered when combining the squares
The number of terms in the expression is minimized
There are no redundant terms (minterms already covered by other terms)

25. Example 3.5 (revisited) Simplify the Boolean Function 00 01 11 10 1 1

26. Example 3.6 (revisited) Simplify the Boolean Function 00 01 11 10 1 1

27. Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1

28. Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1

29. Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1

30. Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1

31. Five-Variable Map

32. Five-Variable Map
How can adjacency be visualized in a five-variable map? 00 01 11 10

33. Five-Variable Map

34. Five-Variable Map
Simplify the Boolean function

35. Product-Of-Sums Simplification
Take the squares with zeros and obtain the simplified complemented function
Complement the above expression and use DeMorgan’s

36. Product-Of-Sums Simplification
Sum-of-products
Product-of-sums

37. Don’t-Care Conditions
Used for incompletely specified functions, e.g. BCD code in which six combinations are not used (1010, 1011, 1100, 1101, 1110, and 1111)
Those unspecified minterms are neither 1’s nor 0’s
Unspecified terms are referred to as “don’t care” and are marked as X
In choosing adjacent squares, don’t care squares can be chosen either as 1’s or 0’s to give the simplest expression

38. Don’t-Care Conditions
Simplify the Boolean function
which has the don’t care conditions

39. NAND and NOR Implementation

40. NAND and NOR Implementation

41. Two-level implementation

42. NAND and NOR Implementation
Implement the following Boolean function with NAND gates

43. Multilevel NAND circuits

44. Multilevel NAND circuits

45. NOR Implementation

46. NOR Implementation

47. NOR Implementation

48. NOR Implementation

49. Exclusive OR Function
Exclusive-OR or XOR performs the following logical operation
Exclusive-NOR or equivalence performs the following logical operation
Identities of the XOR operation
XOR is commutative and associative

50. Exclusive-OR implementations

51. Parity generation and checking

52. Parity generation and checking

53. Parity generation and checking