Boolean Algebra


   
 
Application to Switching Circuits
The application of Boolean algebra to electronic devices such as computers, lies in the restriction of the variable to two possible condition 'On and Off' or 'True or False' or numerically '1 or 0'. The electric circuits carry out the Boolean logic.
 
In practice, electronic engineers use the language of logic as follows.
 
They use the symbol '1' to refer the values of the signals produced by an electronic switch as 'On' or 'True'.
 
They use the symbol '0' to refer the values of the signals produced by an electronic switch as 'Off' or 'False'. The symbols 0 and 1 are called bits.
 
We associate two logical operations 'AND' and 'OR' operations with switching circuits in 'series' and 'parallel' respectively.
 
Logical 'AND' Operation
 
 
Let us refer to a circuit consisting of two switches p and q connected in series with a lamp and battery as shown in figure.
 
The lamp will glow, only if switch p and switch q are closed. If we replace the word 'closed' by T and 'open' by F, the switch will glow only if p = T and q = T. In binary language, we say the switch will glow if p = 1 and q = 1.
 
Table 1, Table 2 and Table 3 describes all possible states of the switches for the series connection.
 
Switches in Series
 
Table 1
 
 
Table 2 - Truth table for p and q
 
 
The 'AND' operation can be defined on the set of bits {0, 1} as follows
 
1 . 1 = 1, 1 . 0 = 0, 0.1 = 0, 0.0 = 0
 
In terms of bits - Table 3
 
 
It is clear that 'AND' operation stipulated that with two input variables, the output is true only when both the inputs are true.
 
Logical 'OR' operation
 
Let us refer to a circuit consisting of two switches p and q connected in parallel with a lamp and battery as shown in figure.
 
 
In this case, the lamp will glow if and only if at least one of the switches is closed. In binary language we say the switch will glow if at least one of the values of p and q is 1.
 
Table 4, Table 5 and Table 6 describe all possibles states of the switches for the 'OR' operation.
 
The 'OR' operation can be defined as the set of bits {0, 1} as follows:
 
1 + 1 = 1, 1 + 0 = 1, 0 + 1 = 1, 0 + 0 = 0
 
Switches in Parallel - Table 4
 
 
Truth Table for p OR q -Table 5
 
 
In terms of bits - Table 6
 
 
The tables such as 3 and 6 are called input/output table with all possible values of inputs in bits and the corresponding possible output in bits.
 
The OR operation stipulates that with two binary input variables, the output is true if either or both the inputs are true.
 
Logical 'NOT' Operation
 
This operation has one input and one output. Table 7 represents truth table for NOT operation.
 
Table 7
 
 
Note that the 'NOT' operation is a unary operator, whereas 'AND' and 'OR' operators are binary operators.
 
We have already discussed what a Boolean function is. Here is a formal definition for Boolean expression and Boolean function.
 
Definition:
 
Let (B, +, ., ', 0, 1) be a Boolean algebra where B is a non-empty set, '+' and '.' are binary operations, ' is a unary operation with two special elements 0 (Zero element) and 1 (unit element). Let x1, x2, x3 ….xn are in B. Then Boolean expression in x1, x2, x3 …. xn are defined recursively as follows:
 
I)
 
0, 1, x1, x2, x3 …. xn are Boolean expressions.
 
II)
 
If x and y are Boolean expressions, then
 
(a) x'
 
(b) x + y (x v y)
 
(c) x . y (x y)
 
are also Boolean expressions.
 
Function which can be obtained from Boolean expressions are called Boolean functions.
 
Note:
 
In the above definition, the symbols 0 and 1 do not necessarily represent the numbers 0 and 1.
 
We can construct an input/output table for a given Boolean function. We can also construct a Boolean function from an input/output table.
 
Example 1:
 
Construct an input/output table for the Boolean function.
 
f(x1, x2, x3) = (x1.x2') + x3
 
Suggested answer:
 
The input/output table is given as follows:
 
Input/output Table for (x1 . x2') + x3 - Table 8
 
 
Example 2:
 
Write the Boolean expression and the Boolean function given by the input/output table as given below:
 
Table 9
 
 
Construct the required function as follows:
 
Step 1:
 
Identify all rows of the table where the output is 1. Note that 1st, 2nd, 3rd and the last row has output 1.
 
Step 2:
 
Form the combination (x1, x2, x3) for the rows identified in step 1.
 
Put xi if xi = 1
 
xi if xi = 0
 
For the
 
1st row, the expression is x1.x2.x3
 
2nd row, the expression is x1.x2.x3'
 
3rd row, the expression is x1, x2'.x3
 
Last row, the expression is x1'.x2'.x3'
 
Step 3:
 
Applying OR to all the combinations obtained in step 3, we have the expression
 
x1 x2 x3 + x1 x2 x3' + x1 x2' x3 + x1' x2' x3'
 
Step 4:
 
The Boolean function can be written as:
 
f(x1, x2, x3) = x1 x2 x3 + x1 x2 x3' + x1 x2' x3 + x1' x2' x3'
 
Since the following is represented by a Boolean expression, it is a Boolean function.
 
  
     
   
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