A maxterm, or

sumterm, is a term in a

logical equation where:

For a function of X input variables, there are 2^{X} maxterms. We represent these as M_{y} where y is the number of the maxterm we are referring to:

For an equation of 3 variables A, B, and C, we number these maxterms as follows:

M_{0} = A + B + C
_
M_{1} = A + B + C
_
M_{2} = A + B + C
_ _
M_{3} = A + B + C
_
M_{4} = A + B + C
_ _
M_{5} = A + B + C
_ _
M_{6} = A + B + C
_ _ _
M_{7} = A + B + C

Notice the order of the complementing. Also notice that it is exactly the **opposite** of minterm numbering. This is a result of DeMorgan's Laws.

Using maxterms we can create a canonical product of sums form to represent any function. We do this by ANDing together all the maxterms, which, when 'input', result in an output of '0' on the function's truth table.

A B C | F
------+--
0 0 0 | 0 <- M_{0}
0 0 1 | 1
0 1 0 | 1
0 1 1 | 0 <- M_{3}
1 0 0 | 1
1 0 1 | 0 <- M_{5}
1 1 0 | 0 <- M_{6}
1 1 1 | 0 <- M_{7}

Which results in the equation: F = M_{0} · M_{3} · M_{5} · M_{6} · M_{7}

Notice that this is the complement of the set of minterms that we used to define this very same function. Also notice that every place a '1' is present in the inputs of this truth table, a complemented variable is used in the maxterm. i.e.:

_ _ _
1 1 1 | 0 <- M_{7} = A + B + C

This is also the opposite of the minterm, where '0's result in complements.

See also Product of Sums.