Boolean Functions

Boolean Functions

Boolean algebra gives a concise way to state the operation of a logic circuit that is produced by a combination of logic gates in order to determine the output based on the combination of various input.

All Boolean expressions can be stated in either two standard forms:

Sum of product OR Product of sum

Sum of product(SOP)

A SOP expression is formed based on the output with the value of 1

Convert each binary value for each product term by replacing each value of 1 with the equivalent variable and each 0 with the equivalent variable complement.

Example:

Product of sum(POS)

A sum term is defined as a term with Boolean additions of variables or their complements.

These sum terms are multiplied to produce a POS expression.

POS expression is formed based on the output with the value of 0.

Convert each binary value to the equivalent sum by substitute each 1 with the equivalent variable complement and each 0 with the equivalent variable.

Example:

Simplification of Combinational Circuits

Examples:

Output =AB+A(B+C)+B(B+C)

            =AB+AB+AC+BB+BC

            =AB+AC+B+BC

            =AB+AC+B(1+C)

            =AB+AC+B

            =AB+B+AC

            =B(A+1)+AC

            =B+AC

Output =(AB+AC)’+A’B’C

            =(AB)’(AC)’+A’B’C

            =(A’+B’)(A’+C’)+A’B’C

            =A’A’+A’B’+A’C’+B’C’+ A’B’C

            =A’+ A’B’+A’C’+B’C’+ A’B’C

            =A’+A’C’+B’C’+ A’B’+ A’B’C

=A’+A’C’+B’C’+ A’B’(1+C)

=A’+A’C’+B’C’+ A’B’

=A’(1+C’)+ B’C’+ A’B’

=A’+B’C’+ A’B’

=A’+A’B’+B’C’

=A’(1+B’)+ B’C’

A’+B’C’