Conversion of Boolean Function - R.D.Sivakumar

1. R.D.SIVAKUMAR, M.Sc.,M.Phil.,M.Tech.,
Assistant Professor of Computer Science &
Assistant Professor and Head, Department of M.Com.(CA),
Ayya Nadar Janaki Ammal College,
Sivakasi – 626 124.
Mobile: 099440-42243
e-mail : sivamsccsit@gmail.com
website: www.rdsivakumar.blogspot.in
Conversion of Boolean Function

2. Conversion of Boolean Function
To build more complicated Boolean functions, we use the AND, OR, and NOT
operators and combine them together. Also, it is possible to convert back and forth
between the three representations of a Boolean function (equation, truth table, and logic
circuit). The following examples show how this is done.
Converting a Boolean Equation to a Truth Table
Truth table lists all the values of the Boolean function for each set of values of the
variables. Now we will obtain a truth table for the following Boolean function
D = (A · B) + C
Clearly, D is a function of three input variables A, B, and C. Hence the truth table will
have 23 = 8 entries, from 000 to 111. Before determining the output D in the table, we
will compute the various intermediate terms like A · B and C as shown in the table
below. For instance, if A = 0, B = 0 and C = 0 then
D = ( A . B ) + C
= ( 0 . 0 ) + 0
= 0 + 0
= 0 + 1
= 1
Here we use the hierarchy of operations of the Boolean operators NOT, AND and OR
over the parenthesis.

3. Conversion of Boolean Function
The truth table for the Boolean function is
Converting a Boolean Equation to a Logic Circuit
The boolean function is realized as a logic circuit by suitably arranging the
logic gates to give the desired output for a given set of input. Any boolean function may
be realized using the three logical operations NOT, AND and OR. Using gates we can
realize Boolean function. Now we will draw the logic circuit for the boolean function.
E = A + ( B · C ) + D
This boolean function has four inputs A, B, C, D and an output E. The output
E is obtained by ORing the individual terms given in the right side of the boolean
function. That is, by ORing the terms A, ( B · C ) and D.

4. Conversion of Boolean Function

5. Conversion of Boolean Function
Converting a Logic Circuit to a Boolean Function
As a reversal operation, the realization of the logic circuit can be expressed as a
boolean function. Let us formulate an expression for the output in terms of the inputs for
the given the logic circuit
To solve this, we simply start from left and work towards the right, identifying and
labeling each of the signals that we encounter until we arrive at the expression for the
output. The labeling of all the signals is shown in the figure below. Let us label the input
signals as A, B, C and the output as D.
Hence the boolean function corresponding to the logic circuit can be written as

6. Conversion of Boolean Function
Converting a Truth Table to a Boolean Function
There are many ways to do this conversion. A simplest way is to write the
boolean function as an OR of minterms. A minterm is simply the ANDing of all
variables, and assigning bars (NOT) to variables whose values are 0.
For example, assuming the inputs to a 4-variable boolean function as A, B, C,
and D the minterm corresponding to the input 1010 is : A • B • C • D. Notice that
this minterm is simply the AND of all four variables, with B and D complemented.
 The reason that B and D are complemented is because for this input, B = D = 0.
As another example, for the input 1110, only D = 0, and so the corresponding
minterm is A • B • C • D.

7. Conversion of Boolean Function
To do this problem, we first circle all of the rows in the truth table which have an
output D = 1. Then for each circled row, we write the corresponding minterm. This
is illustrated in the table below.
Finally, the boolean expression for D is obtained by ORing all of the minterms as
follows:
D = ( A · B · C ) + ( A · B · C ) + ( A · B · C ) + ( A · B · C )

8. Conversion of Boolean Function
Design of Logic Circuit
There are many steps in designing a logic circuit.
First, the problem is stated (in words).
Second, from the word description, the inputs and outputs are identified, and a
block diagram is drawn.
Third, a truth table is formulated which shows the output of the system for every
possible input.
 Fourth, the truth table is converted to a boolean function.
Fifth, the boolean function is converted to a logic circuit diagram.
Finally, the logic circuit is built and tested.
Let us consider the design aspects of a 2-input /single output system which
operates as follows: The output is 1 if and only if precisely one of the inputs is 1;
otherwise, the output is 0.

9. Conversion of Boolean Function
Step 1: Statement of the problem . Given above.
Step 2: Identify inputs and outputs. It is clear from the statement of the problem
that we need two inputs, say A and B, and one output, say C. A block diagram for
this system is
Step 3: Formulate truth table. The truth table for this problem is given below.
Notice that the output is 1 if only one of the inputs is 1. otherwise the output is ‘0’.

10. Conversion of Boolean Function
By ORing the minterms, we obtain the boolean function corresponding to the
truth table as
D = ( A • B ) + ( A • B )
Step 5: Realization of the Boolean function into a Logic Circuit Diagram.
The logic circuit diagram corresponding to this boolean function is given below.