minterms

Minterms are a fundamental concept in Boolean algebra and digital logic. A minterm is a product term (an AND of literals) that corresponds to a single particular combination of input values. By construction, a minterm evaluates to true for exactly one assignment of the variables and false for all other assignments.

For n variables x1, x2, ..., xn, there are 2^n minterms, usually denoted m0 through m_{2^n−1}. Each minterm mi is formed by including the literal xi if the i-th bit of i is 1, and the negated literal ¬xi if that bit is 0, for every variable. For example, with two variables x and y, the minterms are: m0 = ¬x ¬y, m1 = ¬x y, m2 = x ¬y, m3 = x y. Each mi evaluates to 1 only for the specific input combo that corresponds to i’s binary representation.

Any Boolean function F of the variables can be expressed as a sum (OR) of minterms for which F evaluates to 1. This canonical disjunctive normal form is written as F = m_{k1} + m_{k2} + ... + m_{kt}, where the indices ki are precisely the input patterns that yield 1. The complementary canonical form uses maxterms, yielding a product of sums for the patterns that yield 0.

Minterms provide a complete, nonredundant basis for representing Boolean functions and are particularly useful in analysis and design methods such as Karnaugh maps and logic minimization. However, for many variables, the canonical form can become large, so practical implementations often seek simplified expressions.