Boolean function minimization is an essential part of digital logic. K-Map is one of the important methods for Boolean function minimization. In this article, we will learn about why the K-Map uses sequence 00, 01, 11, 10 instead of 00, 01, 10,11. K-Map is the basic method for the Boolean function minimization which eliminates the redundant terms in the function. Let’s start our learning on K-maps and the sequence used in the K-map.
What is Boolean Function Minimization?
Boolean Function minimization is the process of minimizing and simplifying the algebraic expression in the Boolean function so that it can be expressed with the minimum number of literals. The minimization of Boolean functions also reduces the cost and complexity of the circuit. The Boolean functions can be minimized in different ways.
Some of these are:
- Minimization using Boolean algebra
- Minimization using K-map
What is K-Map (Karnaugh’s Map)?
- K-Map is the method used to minimize and simplify the Boolean functions.
- It avoids redundant terms in the Boolean function.
- In K-Map minterms are represented by ∑m and maxterms are represented by πM.
- In K-Map the consecutive cells must have only 1-bit changes.
K-map Examples
Example: Obtain the Boolean function using K-map:
- F(A, B, C, D) = ∑m (1,2,3,4,8,9,12,15)
- F(A, B, C, D) = πM (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
- A group in K-Map contains all 1’s (minterm) or all 0’s (maxterm).
- Group may be horizontal or vertical but not diagonal.
- Groups must contain 2n cells.
- Each group should be as large as possible.
- Each cell containing a one must be in at least one group.
- Groups may overlap and may wrap around the table.
Table for 4-variable binary to gray code
|
Binary Code |
Gray Code |
||||||
|---|---|---|---|---|---|---|---|
|
A |
B |
C |
D |
A |
B |
C |
D |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
|
0 |
0 |
1 |
0 |
0 |
0 |
1 |
1 |
|
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
|
0 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
|
0 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
|
0 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
|
0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
|
1 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
1 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
|
1 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
|
1 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
|
1 |
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
|
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
|
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
Why K-Map has states in sequence of 00, 01, 11, 10 instead of 00, 01, 10, 11?
- The definition of the K-Map states that consecutive cells of the K-Map must have only 1-bit change.
- The sequence 00, 01, 10, 11 is called binary code sequence and the sequence 00, 01, 11, 10 is called the gray code sequence.
- In the binary code sequence 00, 01, 10, 11 the second and third state i.e., 01 and 10 violates the rule that the consecutive cells must have only 1- bit change as in these states there is 2-bit difference. So, the binary sequence 00, 01, 10, 11 cannot be used in K-Map
- The gray code sequence 00, 01, 11, 10 follows the rule that consecutive cells must have only 1-bit as every consecutive state in this sequence has only 1-bit difference. So, it is used in K-Map.
- The above stated reasons the K-Map has states in sequence of 00, 01, 11, 10 instead of 00, 01, 10, 11.
|
For sequence 00, 01, 10, 11 (Binary code) |
For sequence 00, 01, 11, 10 (Gray code) |
|---|---|
|
Consecutive state 01 and 10 has 2- bit difference. 0 MSB of second state and 1 MSB of third state is different as well as 1 LSB of second state and 0 LSB of third state is also different. Hence, 2-bit difference. |
Consecutive states 01 and 11 has 1-bit difference. 0 MSB of second state and 1 MSB of third state is different and LSB of both the states are same. Hence, 1-bit difference. |
|
Second state 0 1 |
Second state 0 1 |
|
Third state 1 0 |
Third state 1 1 |
1-Bit Change Representation in 4-Variable K-map
1-Bit Change Representation in 4-Variable K-map
Applications of K-Map
- K-Map is widely used methods for the minimization of the Boolean functions.
- It avoids the redundant terms in the function.
- It is tabular structure which gives more information about the functions in comparison to the truth table.
Solved Examples on K-Map
Example 1: Minimize the given Boolean function using K-Map: F(A, B, C, D) = ∑m (4, 5, 7, 8, 9, 12, 14)
Solution:
Example 2: Minimize the given Boolean function using K-Map: F(A, B, C, D) = F(A, B, C, D) = πM (0,5,6,7,8,11,15)
Solution:
FAQs on K-map
Q.1: Why we use K-maps?
Answer:
K-maps are used to minimize the Boolean functions.
Q.2: Why gray code is used instead of binary codes in K-map?
Answer:
By the definition of the K-Map, the difference between the consecutive cells of K-map should be 1-bit only. In binary code the difference between cells is 2-bits whereas in gray code the difference between cells is 1-bit. So, gray codes are used instead of binary codes in K-Map.
Q.3: What is the sequence 00, 01, 11, 10 called?
Answer:
The sequence 00, 01, 11, 10 is called gray code. Gray codes are codes used in K-Map.
Q.4: Name the two forms that are used in K-Map.
Answer:
The two forms used in K-map are: SOP (Sum of product) and POS (Product of Sum).