Editing Canonical normal form (section)

== Maxterms {{anchor|Maxterm}} ==
For a [[boolean function]] of {{mvar|n}} variables <math>{x_1,\dots,x_n}</math>, a '''maxterm''' is a sum term in which each of the {{mvar|n}} variables appears ''exactly once'' (either in its complemented or uncomplemented form). Thus, a ''maxterm'' is a logical expression of {{mvar|n}} variables that employs only the complement operator and the disjunction operator ([[logical OR]]). Maxterms are a dual of the minterm idea, following the complementary symmetry of [[De Morgan's laws]]. Instead of using ANDs and complements, we use ORs and complements and proceed similarly. It is apparent that a maxterm gives a ''false'' value for just one combination of the input variables, i.e. it is true at the maximal number of possibilities. For example, the maxterm ''a''&prime; + ''b'' + ''c''&prime; is false only when ''a'' and ''c'' both are true and ''b'' is false—the input arrangement where a = 1, b = 0, c = 1 results in 0.

=== Indexing maxterms ===
There are again 2<sup>''n''</sup> maxterms of {{mvar|n}} variables, since a variable in the maxterm expression can also be in either its direct or its complemented form—two choices per variable. The numbering is chosen so that the complement of a minterm is the respective maxterm. That is, each maxterm is assigned an index based on the opposite conventional binary encoding used for minterms. The maxterm convention assigns the value 0 to the direct form <math>(x_i)</math> and 1 to the complemented form <math>(x'_i)</math>. For example, we assign the index 6 to the maxterm <math>a' + b' + c</math> (110) and denote that maxterm as ''M''<sub>6</sub>. The complement <math>(a' + b' + c)'</math> is the minterm <math>a b c' = m_6</math>, using [[de Morgan's law]].

===Maxterm canonical form===
If one is given a [[truth table]] of a logical function, it is possible to write the function as a "product of sums" or "product of maxterms". This is a special form of [[conjunctive normal form]]. For example, if given the truth table for the carry-out bit ''co'' of one bit position's logic of an adder circuit, as a function of ''x'' and ''y'' from the addends and the carry in, ''ci'':

{| class="wikitable" style="margin: 1em auto 1em auto"
!width="50"|ci
!width="50"|x
!width="50"|y
!width="50"|co(ci,x,y)
|-
|0||0||0||0
|-
|0||0||1||0
|-
|0||1||0||0
|-
|0||1||1||1
|-
|1||0||0||0
|-
|1||0||1||1
|-
|1||1||0||1
|-
|1||1||1||1
|}

Observing that the rows that have an output of 0 are the 1st, 2nd, 3rd, and 5th, we can write ''co'' as a product of maxterms <math>M_0, M_1, M_2</math> and <math>M_4</math>. If we wish to verify this:
:<math>co(ci, x, y) = M_0 M_1 M_2 M_4 = (ci + x + y) (ci + x + y') (ci + x' + y) (ci' + x + y)</math>
evaluated for all 8 combinations of the three variables will match the table.