Page 63 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes For a three-variable expression, the minterms and maxterms are as follows:
Table 4.2: Three-variable Expression of Minterms and Maxterms

X Y Z Minterm Maxterm
0 0 0 X′.Y′.Z′ X+Y+Z
0 0 1 X′.Y′.Z X+Y+Z′
0 1 0 X′.Y.Z′ X+Y′+Z
0 1 1 X′.Y.Z X+Y′+Z′
1 0 0 X.Y′.Z′ X′+Y+Z
1 0 1 X.Y′.Z X′+Y+Z′
1 1 0 X.Y.Z′ X′+Y′+Z
1 1 1 X.Y.Z X′+Y′+Z′

This allows us to represent expressions in either Sum-of-Products or Product-of-Sums forms.

A Boolean expression only composed a combination of the Boolean constants
true or false, Boolean-typed variables, Boolean-valued operators, and Boolean-
valued functions.

4.2 Sum-of-Products and Product-of-Sums Forms of Logic Expressions

Logical functions are expressed in terms of logical variables. The values taken by logical variables
and logical functions are in binary form. Any logical function may he expressed in the following
two forms:
1. Sum-of-products (SOP) form
2. Product-of-sums (POS) form

4.2.1 Sum-of-Products Form

A sum-of-products expression consists of product terms logically added. A product term is a logical
product of many variables. The variables used may or may not be complemented. Following are
the examples of sum of products.
(i) AB + AB + AB
(ii) AB + ABC + BC
(iii) A + AB + BC
(iv) ABC + AB + ABC + ABC

Prepare a truth table for minterms and maxterms.

4.2.2 Product-of-Sums Form

A product-of-sums expression consists of sum terms logically multiplied. A sum term is the logical
addition of several variables. The variables may or may not be complemented.

Following are the examples of product of sums:

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