ONT Re: Differential Logic -- Series B

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DLOG.  Note B17

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Any cell in a venn diagram has a well-defined set of nearest neighbors,
and so we can apply a boundary operator of the appropriate rank to the
list of signed features that conjoined would indicate the cell in view.

For example, having computed the "boundary", or what is more properly
called the "point omitted neighborhood" (PON) of the center cell in a
3-dimensional universe of discourse, what is the PON of the cell that
is furthest from it, namely, the "origin cell" indicated as (x)(y)(z)?

The region bordering the origin cell, (x)(y)(z), can be computed by placing
its three signed conjuncts in a 3-place bracket like (__, __, __), arriving
at the cactus expression that is shown below in both graph and string forms.

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                     o  o  o                     |
|                     |  |  |                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                  ((x),(y),(z))                  |
o-------------------------------------------------o

Figure 11 shows the venn diagram of this expression,
whose meaning is adequately suggested by the phrase
"just 1 of 3 is true".

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o`````````````````````````o                |
|                |``````````` X ```````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\          o          /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /     \    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |``````` Y ```````|       |`````` Z ````````|       |
|       |`````````````````|       |`````````````````|       |
|       o`````````````````o       o`````````````````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 11.  Venn Diagram for ((x),(y),(z))

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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