Canonical and Standard forms

A binary variable may be either in its true form A or its complement Ᾱ. For n variables, the maximum number of input variable combinations is given by N = 2ⁿ. Then considering the AND gate, each of the N logic expressions formed is called a standard product or minterm. As indicated in Table 1, binary digits '1' and '0' are taken to represent a given variable for example A or its complement Ᾱ respectively. Also from Table 1 note that each minterm is assigned a symbol (P j) each where j is the decimal equivalent to the binary number of the minterm designated.

Similarly, if we consider an OR gate, each of the N logic expressions formed is called a standard sum or maxterm. In this case binary digits '1' and '0' are taken to represent a given complemented variable Ᾱ and its true form A respectively. As shown in Table 1, a symbol (S j) is assigned to each maxterm where j is the decimal equivalent to the binary number of the maxterm designated. Also observe that each maxterm is the complement of its corresponding minterm, and vice versa.

Table 1

The minterms and maxterms may be used to define the two standard forms for logic expressions, namely the sum of products (SOP), or sum of minterms, and the product of sums (POS), or product of maxterms. These standard forms of expression aid the logic circuit designer by simplifying the derivation of the function to be implemented. Boolean functions expressed as a sum of products or a product of sums are said to be in canonical form. Note the POS is not the complement of the SOP expression.

SUM OF PRODUCTS (OR of AND terms)

The SOP expression is the equation of the logic function as read off the truth table to specify the input combinations when the output is a logical 1. To illustrate, let us consider Table 2.

Table 2

Observe that the output is high for the rows labelled 3, 5 and 6. The SOP expression for this circuit is thus given any of the following:

 F = P ₃ + P ₅ + P 6

Each product (AND) term is a Minterm. ANDed product of literals in which each variable appears exactly once, in true or complemented form (but not both). Each minterm has exactly one '1' in the truth table. When minterms are ORed together each minterm contributes a '1' to the final function. Note that all product terms are not minterms.

PRODUCT OF SUMS ( AND of OR terms)

The POS expression is the equation of the logic function as read off the truth table to specify the input combinations when the output is a logical 0. To illustrate, let us again consider Table 2. Observe that the output is low for the rows labeled 0, 1, 2, 4 and 7. The POS expression for this circuit is thus given by any of the following:

 F = S₀ S₁ S₂ S₄ S7

Each OR (sum) term is a Maxterm. ORed product of literals in which each variable appears exactly once, in true or complemented form (but not both). Each maxterm has exactly one '0' in the truth table. When maxterms are ANDed together each maxterm contributes a '0' to the final function. Please note that not all sum terms are maxterms.