Editing Propositional formula (section)

==== Minterms ====
In the same way that a 2<sup>n</sup>-row truth table displays the evaluation of a propositional formula for all 2<sup>n</sup> possible values of its variables, n variables produces a 2<sup>n</sup>-square Karnaugh map (even though we cannot draw it in its full-dimensional realization). For example, 3 variables produces 2<sup>3</sup> = 8 rows and 8 Karnaugh squares; 4 variables produces 16 truth-table rows and 16 squares and therefore 16 [[minterms]]. Each Karnaugh-map square and its corresponding truth-table evaluation represents one minterm.

Any propositional formula can be reduced to the "logical sum" (OR) of the active (i.e. "1"- or "T"-valued) minterms. When in this form the formula is said to be in [[disjunctive normal form]]. But even though it is in this form, it is not necessarily minimized with respect to either the number of terms or the number of literals.

In the following table, observe the peculiar numbering of the rows: (0, 1, 3, 2, 6, 7, 5, 4, 0). The first column is the decimal equivalent of the binary equivalent of the digits "cba", in other words:
* Example
*: cba<sub>2</sub> = c*2<sup>2</sup> + b*2<sup>1</sup> + a*2<sup>0</sup>:
*: cba = (c=1, b=0, a=1) = 101<sub>2</sub> = 1*2<sup>2</sup> + 0*2<sup>1</sup> + 1*2<sup>0</sup> = 5<sub>10</sub>

This numbering comes about because as one moves down the table from row to row only one variable at a time changes its value. [[Gray code]] is derived from this notion. This notion can be extended to three and four-dimensional [[hypercube]]s called [[Hasse diagram]]s where each corner's variables change only one at a time as one moves around the edges of the cube. Hasse diagrams (hypercubes) flattened into two dimensions are either [[Veitch diagram]]s or [[Karnaugh map]]s (these are virtually the same thing).

When working with Karnaugh maps one must always keep in mind that the top edge "wrap arounds" to the bottom edge, and the left edge wraps around to the right edge—the Karnaugh diagram is really a three- or four- or n-dimensional flattened object.

{|class="wikitable"
|- style="font-size:9pt" align="center" valign="bottom"
! width="60" Height="39" | decimal equivalent of (c, b, a)
! c
! b
! a
! minterm
|- style="font-size:9pt" align="center" valign="bottom"
| Height="15" | 0
| 0
| 0
| 0
|style="background-color:#FDE9D9" | (~c  &  ~b  &  ~a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="15" | 1
| 0
| 0
| 1
|style="background-color:#FDE9D9" | (~c  &  ~b  &  a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="12" | 3
| 0
| 1
| 1
|style="background-color:#FDE9D9" | (~c  &  b  &  a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="12" | 2
| 0
| 1
| 0
|style="background-color:#FDE9D9" | (~c &  b  &  ~a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="12" | 6
| 1
| 1
| 0
|style="background-color:#FDE9D9" | (c  &  b  &  ~a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="12" | 7
| 1
| 1
| 1
|style="background-color:#FDE9D9" | (c &  b &  a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="12" | 5
| 1
| 0
| 1
|style="background-color:#FDE9D9" | (c  &  ~b  &  a)
|- style="font-size:9pt" align="center" valign="bottom"
| Height="12" | 4
| 1
| 0
| 0
|style="background-color:#FDE9D9" | (c  &  ~b  &  ~a)
|- style="font-size:9pt;color:#A5A5A5" align="center" valign="bottom"
| Height="12" | 0
|style="font-weight:bold" | 0
| 0
| 0
| (~a  &  ~b  &  ~c)
|}