Editing Canonical normal form (section)

===Canonical and non-canonical consequences of NOR gates===
A set of 8 NOR gates, if their inputs are all combinations of the direct and complement forms of the 3 input variables ''ci, x,'' and ''y'', always produce minterms, never maxterms—that is, of the 8 gates required to process all combinations of 3 input variables, only one has the output value 1.  That's because a NOR gate, despite its name, could better be viewed (using De Morgan's law) as the AND of the complements of its input signals.

The reason this is not a problem is the duality of minterms and maxterms, i.e. each maxterm is the complement of the like-indexed minterm, and vice versa.

In the minterm example above, we wrote <math>u(ci, x, y) = m_1 + m_2 + m_4 + m_7</math> but to perform this with a 4-input NOR gate we need to restate it as a product of sums (PoS), where the sums are the opposite maxterms.  That is,

:<math>u(ci, x, y) = \mathrm{AND}(M_0,M_3,M_5,M_6) = \mathrm{NOR}(m_0,m_3,m_5,m_6).</math>
{| style="margin: 1em auto 1em auto"
|+ '''Truth tables'''
|
{| class="wikitable" style="margin: 1em auto 1em auto"
!width="50"|ci
!width="50"|x
!width="50"|y
!width="50"|M<sub>0</sub>
!width="50"|M<sub>3</sub>
!width="50"|M<sub>5</sub>
!width="50"|M<sub>6</sub>
!width="50"|AND
!width="50"|u(ci,x,y)
|-
|0||0||0||0||1||1||1||0||0
|-
|0||0||1||1||1||1||1||1||1
|-
|0||1||0||1||1||1||1||1||1
|-
|0||1||1||1||0||1||1||0||0
|-
|1||0||0||1||1||1||1||1||1
|-
|1||0||1||1||1||0||1||0||0
|-
|1||1||0||1||1||1||0||0||0
|-
|1||1||1||1||1||1||1||1||1
|}
|-
|
{| class="wikitable" style="margin: 1em auto 1em auto"
!width="50"|ci
!width="50"|x
!width="50"|y
!width="50"|m<sub>0</sub>
!width="50"|m<sub>3</sub>
!width="50"|m<sub>5</sub>
!width="50"|m<sub>6</sub>
!width="50"|NOR
!width="50"|u(ci,x,y)
|-
|0||0||0||1||0||0||0||0||0
|-
|0||0||1||0||0||0||0||1||1
|-
|0||1||0||0||0||0||0||1||1
|-
|0||1||1||0||1||0||0||0||0
|-
|1||0||0||0||0||0||0||1||1
|-
|1||0||1||0||0||1||0||0||0
|-
|1||1||0||0||0||0||1||0||0
|-
|1||1||1||0||0||0||0||1||1
|}
|}
In the maxterm example above, we wrote <math>co(ci, x, y) = M_0 M_1 M_2 M_4</math> but to perform this with a 4-input NOR gate we need to notice the equality to the NOR of the same minterms.  That is,

:<math>co(ci, x, y) = \mathrm{AND}(M_0,M_1,M_2,M_4) = \mathrm{NOR}(m_0,m_1,m_2,m_4).</math>

{| style="margin: 1em auto 1em auto"
|+ '''Truth tables'''
|
{| class="wikitable" style="margin: 1em auto 1em auto"
!width="50"|ci
!width="50"|x
!width="50"|y
!width="50"|M<sub>0</sub>
!width="50"|M<sub>1</sub>
!width="50"|M<sub>2</sub>
!width="50"|M<sub>4</sub>
!width="50"|AND
!width="50"|co(ci,x,y)
|-
|0||0||0||0||1||1||1||0||0
|-
|0||0||1||1||0||1||1||0||0
|-
|0||1||0||1||1||0||1||0||0
|-
|0||1||1||1||1||1||1||1||1
|-
|1||0||0||1||1||1||0||0||0
|-
|1||0||1||1||1||1||1||1||1
|-
|1||1||0||1||1||1||1||1||1
|-
|1||1||1||1||1||1||1||1||1
|}
|-
|
{| class="wikitable" style="margin: 1em auto 1em auto"
!width="50"|ci
!width="50"|x
!width="50"|y
!width="50"|m<sub>0</sub>
!width="50"|m<sub>1</sub>
!width="50"|m<sub>2</sub>
!width="50"|m<sub>4</sub>
!width="50"|NOR
!width="50"|co(ci,x,y)
|-
|0||0||0||1||0||0||0||0||0
|-
|0||0||1||0||1||0||0||0||0
|-
|0||1||0||0||0||1||0||0||0
|-
|0||1||1||0||0||0||0||1||1
|-
|1||0||0||0||0||0||1||0||0
|-
|1||0||1||0||0||0||0||1||1
|-
|1||1||0||0||0||0||0||1||1
|-
|1||1||1||0||0||0||0||1||1
|}
|}