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Directory:Jon Awbrey/Papers/Cactus Language (view source)

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To understand what this means in logical terms, it is useful to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the next set of Figures represent the computation of the ''enlarged'' or ''shifted'' proposition <math>\operatorname{E}f</math> at each of the 4 points in the universe of discourse <math>U = X \times Y.</math>
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{| align="center" cellpadding="6" width="90%"
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| align="center" |
 
<pre>
 
<pre>
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To understand what this means in logical terms, for instance, as expressed
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in a boolean expansion or a "disjunctive normal form" (DNF), it is perhaps
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a little better to go back and analyze the expression the same way that we
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did for Df.  Thus, let us compute the value of the enlarged proposition Ef
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at each of the points in the universe of discourse U = X x Y.
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o---------------------------------------o
 
o---------------------------------------o
 
|                                      |
 
|                                      |
Line 3,727: Line 3,725:  
| Ef =      (x, dx)·(y, dy)            |
 
| Ef =      (x, dx)·(y, dy)            |
 
o---------------------------------------o
 
o---------------------------------------o
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</pre>
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|-
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| align="center" |
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<pre>
 
o---------------------------------------o
 
o---------------------------------------o
 
|                                      |
 
|                                      |
Line 3,740: Line 3,741:  
| Ef|xy =      (dx)·(dy)              |
 
| Ef|xy =      (dx)·(dy)              |
 
o---------------------------------------o
 
o---------------------------------------o
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</pre>
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|-
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| align="center" |
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<pre>
 
o---------------------------------------o
 
o---------------------------------------o
 
|                                      |
 
|                                      |
Line 3,754: Line 3,758:  
| Ef|x(y) =    (dx)· dy                |
 
| Ef|x(y) =    (dx)· dy                |
 
o---------------------------------------o
 
o---------------------------------------o
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</pre>
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|-
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| align="center" |
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<pre>
 
o---------------------------------------o
 
o---------------------------------------o
 
|                                      |
 
|                                      |
Line 3,768: Line 3,775:  
| Ef|(x)y =      dx ·(dy)              |
 
| Ef|(x)y =      dx ·(dy)              |
 
o---------------------------------------o
 
o---------------------------------------o
−
 
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</pre>
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|-
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| align="center" |
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<pre>
 
o---------------------------------------o
 
o---------------------------------------o
 
|                                      |
 
|                                      |
Line 3,782: Line 3,792:  
| Ef|(x)(y) =    dx · dy                |
 
| Ef|(x)(y) =    dx · dy                |
 
o---------------------------------------o
 
o---------------------------------------o
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</pre>
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<pre>
 
Given the sort of data that arises from this form of analysis,
 
Given the sort of data that arises from this form of analysis,
 
we can now fold the disjoined ingredients back into a boolean
 
we can now fold the disjoined ingredients back into a boolean
Jon Awbrey
12,122

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