Difference between revisions of "User:Jon Awbrey/GRAPHICS"
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Revision as of 12:52, 30 December 2009
Image Gallery
Reduction 6:1
Cacti
| Image | Scale |
 |
117 px → 20 px |
 |
117 px → 20 px |
Riffs
| Image | Scale |
 |
117 px → 20 px |
 |
240 px → 40 px |
 |
233 px → 40 px |
 |
384 px → 65 px |
 |
384 px → 65 px |
 |
377 px → 65 px |
 |
384 px → 65 px |
 |
240 px → 40 px |
Rotes
| Image | Scale |
 |
117 px → 20 px |
 |
233 px → 40 px |
 |
233 px → 40 px |
 |
377 px → 65 px |
 |
233 px → 40 px |
 |
473 px → 80 px |
 |
377 px → 65 px |
 |
377 px → 65 px |
 |
473 px → 80 px |
 |
473 px → 80 px |
 |
233 px → 40 px |
 |
617 px → 105 px |
 |
473 px → 80 px |
 |
617 px → 105 px |
 |
473 px → 80 px |
 |
521 px → 90 px |
 |
377 px → 65 px |
 |
713 px → 120 px |
 |
377 px → 65 px |
 |
617 px → 105 px |
 |
617 px → 105 px |
 |
473 px → 80 px |
 |
473 px → 80 px |
 |
617 px → 105 px |
 |
473 px → 80 px |
 |
713 px → 120 px |
 |
473 px → 80 px |
 |
761 px → 130 px |
 |
473 px → 80 px |
 |
713 px → 120 px |
 |
233 px → 40 px |
 |
377 px → 65 px |
 |
473 px → 80 px |
 |
617 px → 105 px |
 |
617 px → 105 px |
 |
857 px → 145 px |
 |
617 px → 105 px |
 |
617 px → 105 px |
 |
713 px → 120 px |
 |
617 px → 105 px |
 |
473 px → 80 px |
 |
857 px → 145 px |
 |
617 px → 105 px |
 |
617 px → 105 px |
 |
713 px → 120 px |
 |
713 px → 120 px |
 |
473 px → 80 px |
 |
617 px → 105 px |
 |
473 px → 80 px |
 |
713 px → 120 px |
 |
617 px → 105 px |
 |
857 px → 145 px |
 |
521 px → 90 px |
 |
713 px → 120 px |
 |
473 px → 80 px |
 |
761 px → 130 px |
 |
617 px → 105 px |
 |
713 px → 120 px |
 |
377 px → 65 px |
 |
905 px → 155 px |
 |
1001 px → 170 px |
Reduction 10:1
Cacti
| Image | Scale |
|
117 px → 12 px |
|
117 px → 12 px |
Riffs
| Image | Scale |
|
117 px → 12 px |
|
240 px → 24 px |
|
233 px → 24 px |
|
384 px → 38 px |
|
384 px → 38 px |
|
377 px → 38 px |
|
384 px → 38 px |
|
240 px → 24 px |
Rotes
| Image | Scale |
|
117 px → 12 px |
|
233 px → 24 px |
|
233 px → 24 px |
|
377 px → 38 px |
|
233 px → 24 px |
|
473 px → 48 px |
|
377 px → 38 px |
|
377 px → 38 px |
|
473 px → 48 px |
|
473 px → 48 px |
|
233 px → 24 px |
|
617 px → 62 px |
|
473 px → 48 px |
|
617 px → 62 px |
|
473 px → 48 px |
|
521 px → 52 px |
|
377 px → 38 px |
|
713 px → 72 px |
|
377 px → 38 px |
|
617 px → 62 px |
|
617 px → 62 px |
|
473 px → 48 px |
|
473 px → 48 px |
|
617 px → 62 px |
|
473 px → 48 px |
|
713 px → 72 px |
|
473 px → 48 px |
|
761 px → 76 px |
|
473 px → 48 px |
|
713 px → 72 px |
|
233 px → 24 px |
|
377 px → 38 px |
|
473 px → 48 px |
|
617 px → 62 px |
|
617 px → 62 px |
|
857 px → 86 px |
|
617 px → 62 px |
|
617 px → 62 px |
|
713 px → 72 px |
|
617 px → 62 px |
|
473 px → 48 px |
|
857 px → 86 px |
|
617 px → 62 px |
|
617 px → 62 px |
|
713 px → 72 px |
|
713 px → 72 px |
|
473 px → 48 px |
|
617 px → 62 px |
|
473 px → 48 px |
|
713 px → 72 px |
|
617 px → 62 px |
|
857 px → 86 px |
|
521 px → 52 px |
|
713 px → 72 px |
|
473 px → 48 px |
|
761 px → 76 px |
|
617 px → 62 px |
|
713 px → 72 px |
|
377 px → 38 px |
|
905 px → 90 px |
|
1001 px → 100 px |
Rotes 1 to 60
|
\(\begin{array}{l} \varnothing \\ 1 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\) |
\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\) |
\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\) |
|
\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\) |
\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\) |
\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\) |
\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\) |
|
\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\) |
\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\) |
\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\) |
\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\) |
\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\) |
|
\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\) |
\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\) |
\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\) |
|
\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\) |
\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\) |
\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\) |
\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\) |
\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\) |
\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\) |
|
\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\) |
\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\) |
\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\) |
Cactus Graphs
Hi Res
|
|
117 px → 20 px |
|
|
117 px → 20 px |
|
117 px → 20 px |
_Big.jpg){kind=small}
|
117 px → 20 px |
|
290 px → 50 px |
(B)(C))_Big.jpg)
|
386 px → 65 px |
B_Big.jpg){kind=small}
|
204 px → 35 px |
)_Big.jpg)
|
348 px → 60 px |
_Big.jpg)
|
386 px → 65 px |
)_Big.jpg)
|
386 px → 65 px |
_Big.jpg)
|
386 px → 65 px |
,(B),(C))_Big.jpg)
|
386 px → 65 px |
)_Big.jpg)
|
386 px → 65 px |
,(B),(C)))_Big.jpg)
|
386 px → 65 px |
,(C))_Big.jpg)
|
386 px → 65 px |
,B,C))_Big.jpg)
|
386 px → 65 px |
)_Big.jpg)
|
530 px → 90 px |
,(B),(C))_Big.jpg)
|
530 px → 90 px |
Lo Res
|
|
|
|
Differential Logic
ASCII Graphics
Series 1
o-------------------------------------------------o | | | | | o-------------o o-------------o | | / \ / \ | | / o \ | | / /%\ \ | | / /%%%\ \ | | o o%%%%%o o | | | |%%%%%| | | | | P |%%%%%| Q | | | | |%%%%%| | | | o o%%%%%o o | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o | f = p q | o-------------------------------------------------o Figure 22-a. Conjunction pq : X -> B |
o-------------------------------------------------o | | | | | o-------------o o-------------o | | / \ / \ | | / P o Q \ | | / /%\ \ | | / /%%%\ \ | | o o.->-.o o | | | p(q)(dp)dq |%\%/%| (p)q dp(dq) | | | | o---------------|->o<-|---------------o | | | | |%%^%%| | | | o o%%|%%o o | | \ \%|%/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | | | | o | | (p)(q) dp dq | | | o-------------------------------------------------o | f = p q | o-------------------------------------------------o | | | Ef = p q (dp)(dq) | | | | + p (q) (dp) dq | | | | + (p) q dp (dq) | | | | + (p)(q) dp dq | | | o-------------------------------------------------o Figure 22-b. Enlargement E[pq] : EX -> B |
o-------------------------------------------------o | | | | | o-------------o o-------------o | | / \ / \ | | / P o Q \ | | / /%\ \ | | / /%%%\ \ | | o o%%%%%o o | | | (dp)dq |%%%%%| dp(dq) | | | | o<--------------|->o<-|-------------->o | | | | |%%^%%| | | | o o%%|%%o o | | \ \%|%/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | v | | o | | dp dq | | | o-------------------------------------------------o | f = p q | o-------------------------------------------------o | | | Df = p q ((dp)(dq)) | | | | + p (q) (dp) dq | | | | + (p) q dp (dq) | | | | + (p)(q) dp dq | | | o-------------------------------------------------o Figure 22-c. Difference D[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | | | | | | | | | | | | | G | | | | | | | | | | | | | | | o o | | \ / | | \ / | | \ T / | | \ o<------------/-------------o | | \ / | | \ / | | \ / | | o-------------------o | | | | | o---------------------------------------------------------------------o Figure 23. Elements of a Cybernetic System |
Series 2
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / /%\ \ | | / /%%%\ \ | | / /%%%%%\ \ | | / /%%%%%%%\ \ | | / /%%%%%%%%%\ \ | | o o%%%%%%%%%%%o o | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | | P |%%%%%%%%%%%| Q | | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | o o%%%%%%%%%%%o o | | \ \%%%%%%%%%/ / | | \ \%%%%%%%/ / | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 24-1. Proposition pq : X -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (dp) (dq) o o | | | | o-->--o | | | | | | \ / | | | | | (dp) dq | \ / | dp (dq) | | | | o<-----------------o----------------->o | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 24-2. Tacit Extension !e![pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (dp) (dq) o o | | | | o-->--o | | | | | | \ / | | | | | (dp) dq | \ / | dp (dq) | | | | o----------------->o<-----------------o | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | | | | o | | | o---------------------------------------------------------------------o Figure 25-1. Enlargement E[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | (dp) dq | | dp (dq) | | | | o<---------------->o<---------------->o | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 25-2. Difference Map D[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / o \ \ | | / / ^ ^ \ \ | | o o / \ o o | | | | / \ | | | | | | / \ | | | | | |/ \| | | | | (dp)/ dq dp \(dq) | | | | /| |\ | | | | / | | \ | | | | / | | \ | | | o / o o \ o | | \ v \ dp dq / v / | | \ o<--------------------->o / | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 26-1. Differential or Tangent d[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | | dp dq | | | | | o<------------------------------->o | | | | | | | | | | | | | | | | | o | | | | o o ^ o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ dp | dq / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | | | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 26-2. Remainder r[pq] : EX -> B |
JPEG Graphics
Series 1
|
| \(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\) |
|
| \(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) |
|
\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\) |
|
| \(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\) |
|
\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\) |
Series 2
|
|
| \(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\) |
|
| \(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\) |
|
\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\) |
|
|
| \(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) |
|
\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\) |
|
|
| \(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\) |
|
\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\) |
|
| \(\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}\) |
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| \(\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{r}(pq) & = & p & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \end{array}\) |
Propositional Equation Reasoning Systems
Analysis of contingent propositions
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| \(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}\) |
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| \(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}\) |
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Equation 1 : Proof 1
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Equation 1 : Proof 2
Single Image Version
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Serial Image Version
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Equation 1 : Proof 3
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Variant 1
o-----------------------------------------------------------o | Equation E_1. Proof 3. | o-----------------------------------------------------------o | 1 | | q o o r q o r | | | | | | | p o o p p o | | \ / | | | o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | | | | | | | | | | | | @ | | | o==================================< CAST "p" >=============o | 2 | | q r q r q r qr | | o o o o o o o o o | | | | | |/ |/ |/ | | o o o o o o | | \ / | \ / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | p o---------------o---o p | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o==================================< Domination >===========o | 3 | | q r q r | | o o o o o o | | | | | / / / | | o o o o o o | | \ / | \ / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | p o---------------o---o p | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o==================================< Cancellation >=========o | 4 | | q r q r | | o o o | | | | | | | o o o | | \ / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | p o---------------o---o p | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o==================================< Emptiness >============o | 5 | | q r q r | | o o o | | | | | | | o o o | | \ / | | | o-------o o | | \ / | | | \ / | | | \ / | | | o o | | | | | | | | | | | | | | p o---------------o---o p | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o==================================< Cancellation >=========o | 6 | | q r q r | | o o o | | | | | | | o o o | | \ / | | | o-------o | | \ / | | \ / | | \ / | | o | | | | | | | | | | | p o---------------o---o p | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o==================================< CAST "q" >=============o | 7 | | o o | | r r | r | | | o o o o o o r | | | | | | | | | | o o o o o o | | \ / | \ / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | q o---------------o---o q | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Domination >===========o | 8 | | o o | | r r | r | | | o o o o o o | | | | | | | | | | o o o o o o | | \ / | \ / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | q o---------------o---o q | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Cancellation >=========o | 9 | | r r r | | o o o | | | | | | | o o o o o | | / | \ / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | q o---------------o---o q | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Domination >===========o | 10 | | r r | | o o | | | | | | o o o o | | / | \ | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | q o---------------o---o q | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Spike >================o | 11 | | r r | | o o | | | | | | o o | | / | | | o-------o o | | \ / | | | \ / | | | \ / | | | o o | | | | | | | | | | | | | | q o---------------o---o q | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Cancellation >=========o | 12 | | r r | | o o | | | | | | o o | | / | | | o-------o | | \ / | | \ / | | \ / | | o | | | | | | | | | | | q o---------------o---o q | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< CAST "r" >=============o | 13 | | o o | | | | | | o o o o | | | | | | | | o o o o | | / | / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | r o---------------o---o r | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | q o-------o---o q | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Cancellation >=========o | 14 | | o o | | / | | | o-------o o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | | | | | | | | | | | | r o---------------o---o r | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | q o-------o---o q | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Emptiness & Spike >====o | 15 | | o o 16 | | | | | | | | | | | | | | o o | | | | | | | | | | | | | | r o---------------o---o r | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | q o-------o---o q | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< Cancellation >=========o | 17 | | r o-------o---o r | | \ / | | \ / | | \ / | | q o-------o---o q | | \ / | | \ / | | \ / | | p o-------o---o p | | \ / | | \ / | | \ / | | @ | | | o==================================< QED >==================o |
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Variant 2
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Praeclarum Theorema : Proof by CAST
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