Free Courses : Rules And Properties Of KARNAUGH Table

KARNAUGH Table

We have seen the rules and properties of Boolean algebra to simplify functions ; However, this method is relatively cumbersome and never indicates whether it results in minimal expression of the function or not .

We can use the method table of Karnaugh .
In the case of two binary variables, we have four possibilities (or combinations ) to consider that we translate as the following truth table :

For each combination of variables is associated a value of the function.

The idea is to associate a KARNAUGH surface at each combination of variables, by adopting the following representation :

We have 4 boxes of the 4 combinations of variables .
                                                               
Box 1 is the combination a = 0 b = 0 ==> ( a. b)
                                                               
Box 2 is the combination a = 1 b = 0 ==> ( a. b)
                                                            
Box 3 is the combination a = 0 b = 1 ==> ( a. b)

Box 4 is the combination a = 1 b = 1 ==> ( a. b)

In each of these cases will be included the value of the function for the combination of variables corresponding to this box.
Following the example already shown above we have:
box 2 ==> combination of variables a = 1 and b = 0 ==> value of the function = 0.

For each box we associate a product of variables

Representation of a table of Karnaugh

A table of Karnaugh can be represented in the following forms:

These three representations are equivalent.
A table of Karnaugh therefore provides information on the following data:

• The name of the function (eg, X )
• The names of variables (a, b )
• The state variables : 0, 1 or bar representing the state 1
• The function value ( 1 or 0) .

We note that :
In box 1 the variables are all equal to 0.
If the Boolean algebraic notation is adopted for the variables, it informs us of the name and the state of the variable (a,!a).

Table 3 Variable Karnaugh

Each box is associated with a triplet of the values ​​a , b , c .
Example : The box 1 represent the triplet { 0 ,0 ,0 }  :  a = 0, b = 0 and c = 0.                                                                                                                  
We can also say that the box 1 is the product ( a . b . c).
In this case the representation becomes:

Table 4 variable Karnaugh

Each box is associated with a tuple of values ​​a, b ​​, c , d.
Examples:
                                                                                                                                  
The box  4 represent the quadruplet { 1 , 0 , 0 , 0 } : a = 1, b = 0, c = 0 and d = 0 (a . b . c . d).

The box 11 represent the quadruplet { 1 , 1 , 1 , 1 } : a = 1, b = 1 , c = 1 and d = 1 (a . b . c . d).
                                                                                                                                                                   
The box 16 represent the quadruplet { 1 , 0 , 1 , 0 } : a = 1, b = 0, c = 1 and d = 0 (a . b . c . d).

Adjacencies boxes

In each case, the order of writing the variable states that between two neighboring cells ( in a row or column ) one variable changes state ; such cases we say they are adjacent.

The Box 2 corresponds to a = 0 ; b = 1 ; c = 0 ; d = 0
The box 3 corresponds to a = 1 ; b = 1 ; c = 0 ; d = 0
When we move from 2 and 3, only the variable "a" change of state : 2 and 3 are adjacents .
When we move from 2 to 1, only the variable "b" changes state :
1 and 2 are adjacents.
When we move from 2 to 6, only the variable "d" changes state :
2 and 6 are adjacents.
Finally, when we move from 2-14 , only the variable "c" changes state :
2 and 14 are adjacents.
We have to determine the adjacencies of the box 2.

This concept is fundamental to adjacent squares.

See more : Write a function in a table of Karnaugh.  soon