Minterm

A minterm is a product term used in boolean algebra that corresponds to a single assignment of input variables. It is constructed so that the minterm evaluates to 1 for that specific input combination and to 0 for all other combinations.

Formally, for n variables x1, …, xn, there are 2^n minterms m0, m1, …, m(2^n−1). The i-th minterm is the conjunction of literals where the j-th variable is complemented if the j-th bit of i in binary is 0 and uncomplemented if that bit is 1. In this way each minterm represents exactly one row of a truth table.

Any boolean function f of n variables can be expressed as a sum of minterms corresponding to the input patterns for which f equals 1. This canonical disjunctive normal form is written as f = Σ m_i. The complementary canonical form uses maxterms for the input patterns where f equals 0. Minterms therefore provide a standard basis for representing boolean functions.

Example: with two variables x and y, the minterms are m0 = x' y', m1 = x y', m2 = x' y, m3 = x y. A function f that is true for inputs 01 and 10 can be written as f = m1 + m2, which is XOR. Conversely, f = m0 + m3 represents XNOR. Minterms also correspond to cells in Karnaugh maps and underpin systematic methods for logic simplification.