Minterms and maxterms are fundamental concepts in Boolean algebra. A minterm is a product term where each variable appears exactly once, evaluating to 1 for one specific input combination. A maxterm is a sum term where each variable appears exactly once, evaluating to 0 for one specific input combination. Let's examine this truth table example.
A minterm is a product term where each variable appears exactly once, either in true or complemented form. For two variables A and B, there are four possible minterms. Each minterm corresponds to exactly one row in the truth table and evaluates to 1 only for that specific input combination. For example, m₀ equals A prime B prime and is true only when both A and B are zero.
A maxterm is a sum term where each variable appears exactly once, either in true or complemented form. For two variables A and B, there are four possible maxterms. Each maxterm corresponds to exactly one row in the truth table and evaluates to 0 only for that specific input combination. For example, M₁ equals A plus B prime and is false only when A is zero and B is one.
Boolean functions can be expressed in two standard forms. Sum of Products or SOP form uses the OR of all minterms where the function output is 1. Product of Sums or POS form uses the AND of all maxterms where the function output is 0. For our example, the SOP form is m₀ plus m₃, which equals A prime B prime plus AB. The POS form is M₁ times M₂, which equals A plus B prime times A prime plus B.
The duality principle shows that the complement of any minterm equals the corresponding maxterm with the same index. This relationship is fundamental in Boolean algebra. For n variables, there are 2 to the power n minterms and maxterms, each with a unique index. These concepts are essential in digital circuit design, logic simplification, and Boolean algebra applications. Understanding minterms and maxterms provides the foundation for analyzing and designing digital systems.