Re: Dick Lipton & Ken Regan • (1) • (2)
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(3) |
Figure 3 shows the eight terms of the enlarged proposition 
For ease of reference, here is the expansion of
![\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}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bsmallmatrix%7D++%26+p+q+r+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+p+q+%5Ctexttt%7B%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dr++%26+%2B+%26+p+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D+r+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+p+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dq+%5C%2C+%5Cmathrm%7Bd%7Dr++%5C%5C%5B4pt%5D++%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+q+r+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+q+%5Ctexttt%7B%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dr++%26+%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%28%7D+q+%5Ctexttt%7B%29%7D+r+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5C%2C+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%28%7D+q+%5Ctexttt%7B%29%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5C%2C+%5Cmathrm%7Bd%7Dq+%5C%2C+%5Cmathrm%7Bd%7Dr++%5Cend%7Bsmallmatrix%7D&bg=ffffff&fg=000000&s=2&c=20201002)
Two examples suffice to convey the general idea of the enlarged venn diagram:
- The term
is shown as a looped arc starting in the cell where
is true and returning back to it. The differential factor
corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
- The term
is shown as an arc from the cell where
is true to the cell where
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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