Frankl, My Dear : 8

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


Venn Diagram Frankl Figure 4
(4)

Figure 4 shows the eight terms of the tacit extension \boldsymbol\varepsilon f as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr. Each term of the tacit extension \boldsymbol\varepsilon f corresponds to an arc that starts from the cell where f is true and ends in one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of \boldsymbol\varepsilon f from the previous post:

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

\begin{array}{*{8}{l}}  &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  & + &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  \\[4pt]  + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  & + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  \end{array}

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

  • The term pqr \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 pqr 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 pqr \cdot \mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r is shown as an arc going from the cell where pqr is true to the cell where \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} 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 …

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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 : 8

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

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

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