Minterms and Maxterms
Minterms and maxterms are the canonical product and sum terms that pin a Boolean function to individual truth-table rows.
Definitions
A minterm is a product (AND) of every input variable, each plain or complemented, that is 1 for exactly one input combination. A maxterm is a sum (OR) of every input variable that is 0 for exactly one combination. Each row of a truth table has exactly one minterm and one maxterm.
Numbering
Terms are numbered by the binary value of the input combination they single out. For inputs A, B with A the high bit, the row A=1, B=0 is combination 2, so its minterm is m2 = A AND NOT B and its maxterm is M2 = NOT A OR B.
Building functions
- A function is the OR of the minterms of its 1-rows; this is the canonical sum-of-products.
- The same function is the AND of the maxterms of its 0-rows; this is the canonical product-of-sums.
- Minterm and maxterm indices for one function partition the full set of rows.
- Shorthand notation lists the indices, such as F = sum of m(1, 2).
Duality
Minterm mi and maxterm Mi are complements: Mi = NOT(mi). This is a direct consequence of De Morgan's laws and explains why the SOP and POS forms describe the same function from opposite directions.
Why they matter
Canonical terms give an unambiguous algebraic description straight from a truth table, before any minimization. They are the entry point for simplification with algebra or a Karnaugh map, and the standard input format for logic-synthesis tools.