Introduction to Digital Systems: Modeling, Synthesis, and Simulation Using VHDL

3.11 PRODUCT-OF-SUMS DESIGN

Contrary to minterms, which represent the product of variables that set the function to 1, the function can also be represented by the sum of variables, which set the function to 0. Variables used to represent the function using the complement to minterms are called maxterms. All possible maxterms for three-variable functions are listed in Figure 3.9. Consider the function specified by a truth table in Figure 3.10; its complement function can be represented by a sum of minterms for which the function is equal to 0. For example, the complement of function f can be represented as

Using DeMorgan's theorem, the function f can be represented as

Using Boolean algebra identities, the function can be reduced:

Using the shorthand method to express the function for the product of sums yields

The symbol ͠ represents the logical product (AND) operation. More algebraic manipulations and simplifications are explored in the Karnaugh mapping sections in Chapter 6.