Editing Canonical normal form (section)

==Minterms {{anchor|Minterm}}==
For a [[boolean function]] of <math>n</math> variables <math>{x_1,\dots,x_n}</math>, a '''minterm''' is a [[product term]] in which each of the <math>n</math> variables appears ''exactly once'' (either in its complemented or uncomplemented form). Thus, a ''minterm'' is a logical expression of ''n'' variables that employs only the complement operator and the conjunction operator ([[logical AND]]). A minterm gives a true value for just one combination of the input variables, the minimum nontrivial amount. For example, ''a'' ''b''<nowiki>'</nowiki> ''c'', is true only when ''a'' and ''c'' both are true and ''b'' is false—the input arrangement where ''a'' = 1, ''b'' = 0, ''c'' = 1 results in 1.

=== Indexing minterms ===
There are 2<sup>''n''</sup> minterms of ''n'' variables, since a variable in the minterm expression can be in either its direct or its complemented form—two choices per variable. Minterms are often numbered by a binary encoding of the complementation pattern of the variables, where the variables are written in a standard order, usually alphabetical. This convention assigns the value 1 to the direct form (<math>x_i</math>) and 0 to the complemented form (<math>x'_i</math>); the minterm is then <math>\sum\limits_{i=1}^n2^{i-1}\operatorname{value}(x_i)</math>. For example, minterm <math>a b c'</math> is numbered 110<sub>2</sub>&nbsp;=&nbsp;6<sub>10</sub> and denoted <math>m_6</math>.

===Minterm canonical form===

Given the [[truth table]] of a logical function, it is possible to write the function as a "sum of products" or "sum of minterms". This is a special form of [[disjunctive normal form]]. For example, if given the truth table for the arithmetic sum bit ''u'' 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"|u(ci,x,y)
|-
|0||0||0||0
|-
|0||0||1||1
|-
|0||1||0||1
|-
|0||1||1||0
|-
|1||0||0||1
|-
|1||0||1||0
|-
|1||1||0||0
|-
|1||1||1||1
|}

Observing that the rows that have an output of 1 are the 2nd, 3rd, 5th, and 8th, we can write ''u'' as a sum of minterms <math>m_1, m_2, m_4,</math> and <math>m_7</math>. If we wish to verify this: <math> u(ci,x,y) = m_1 + m_2 + m_4 + m_7 = (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.