Frankl, My Dear : 6

Re: Dick Lipton & Ken Regan(1)(2)


Venn Diagram Frankl Figure 3
(3)

Figure 3 shows the eight terms of the enlarged proposition \mathrm{E}f as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr. Each term of the enlargement \mathrm{E}f corresponds to an arc into the cell where f is true from one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of \mathrm{E}f from the previous post:

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{smallmatrix}  & p q r \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + & p q \texttt{(} r \texttt{)} \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r  & + & p \texttt{(} q \texttt{)} r \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}  & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r  \\[4pt]  + & \texttt{(} p \texttt{)} q r \,\cdot\,  \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\,  \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r  & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\,  \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}  & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\,  \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r  \end{smallmatrix}

Two examples suffice to convey the general idea of the enlarged venn diagram:

  • The term p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} is shown as a looped arc starting in the cell where p q r is true and returning back to it. The differential factor \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
  • The term \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r is shown as an arc from the cell where \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} is true to the cell where p q r is true. The differential factor \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

To be continued …

Resources

This entry was posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl and tagged , , , , , , , , . Bookmark the permalink.

2 Responses to Frankl, My Dear : 6

  1. Pingback: Frankl, My Dear : 9 | Inquiry Into Inquiry

  2. Pingback: Survey of Differential Logic • 1 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s