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Chapter: Digital Logic Circuits - Number Systems and Digital Logic Families

Min Terms and Max Terms

Any boolean expression may be expressed in terms of either minterms or maxterms.

MIN TERMS AND MAX TERMS

 

Any boolean expression may be expressed in terms of either minterms or maxterms. To do this we must first define the concept of a literal. A literal is a single variable within a term which may or may not be complemented. For an expression with N variables, minterms and maxterms are defined as follows :

 

A minterm is the product of N distinct literals where each literal occurs exactly once.

 

            A maxterm is the sum of N distinct literals where each literal occurs exactly once.

 

Product-of-Sums Expressions

 

Standard Forms

 

A product-of-sums expression contains the product of different terms, with each term being either a single literal or a sum of more than one literal. It can be obtained from the truth table by considering those input combinations that produce a logic 0 at the output. Each such input combination gives a term, and the product of all such terms gives the expression. Different terms are obtained by taking the sum of the corresponding literals. Here, 0 and ‘1‘ respectively mean the uncomplemented and complemented variables, unlike sum-of-products expressions where ‘0‘ and ‘1‘ respectively mean complemented and uncomplemented variables.

 

Since each term in the case of the product-of-sums expression is going to be the sum of literals, this implies that it is going to be implemented using an OR operation. Now, an OR gate produces a logic ‘0‘ only when all its inputs are in the logic ‘0‘ state, which means that the first term corresponding to the second row of the truth table will be A+B+C. The product-of-sums Boolean expression for this truth table is given by Transforming the given product-of-sums expression into an equivalent sum-of-products expression is a straightforward process. Multiplying out the given expression and carrying out the obvious simplification provides the equivalent sum-of-products expression:

 

A given sum-of-products expression can be transformed into an equivalent product-of-sums expression by (a) taking the dual of the given expression, (b) multiplying out different terms to get the sum-of products form, (c) removing redundancy and (d) taking a dual to get the equivalent product-of-sums expression. As an illustration, let us find the equivalent product-of-sums expression of the sum-of products expression




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