12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Tuesday April 30, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 01:02, 29 May 2009
, 01:02, 29 May 2009→Note 3: markup
In the universe <math>U = X \times Y,</math> the four propositions <math>xy,~ x\texttt{(}y\texttt{)},~ \texttt{(}x\texttt{)}y,~ \texttt{(}x\texttt{)(}y\texttt{)}</math> that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!</math> respectively.
In the universe <math>U = X \times Y,</math> the four propositions <math>xy,~ x\texttt{(}y\texttt{)},~ \texttt{(}x\texttt{)}y,~ \texttt{(}x\texttt{)(}y\texttt{)}</math> that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!</math> respectively.
Thus we can write <math>\operatorname{D}f_p = \operatorname{D}f|p = \operatorname{D}f|(1, 1) = \operatorname{D}f|xy,</math> so long as we know the frame of reference in force.
Thus, we can write Df_p = Df|p = Df|<1, 1> = Df|xy,
so long as we know the frame of reference in force.
Sticking with the example f(x, y) = xy, let us compute the
Sticking with the example <math>f(x, y) = xy,\!</math> let us compute the value of the difference proposition <math>\operatorname{D}f</math> at all 4 points.
value of the difference proposition Df at all of the points.
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| Df = ((x, dx)(y, dy), xy) |
| Df = ((x, dx)(y, dy), xy) |
o---------------------------------------o
o---------------------------------------o
</pre>
|-
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| Df|xy = ((dx)(dy)) |
| Df|xy = ((dx)(dy)) |
o---------------------------------------o
o---------------------------------------o
</pre>
|-
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| Df|x(y) = (dx) dy |
| Df|x(y) = (dx) dy |
o---------------------------------------o
o---------------------------------------o
</pre>
|-
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| Df|(x)y = dx (dy) |
| Df|(x)y = dx (dy) |
o---------------------------------------o
o---------------------------------------o
</pre>
|-
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| Df|(x)(y) = dx dy |
| Df|(x)(y) = dx dy |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
The easy way to visualize the values of these graphical
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
expressions is just to notice the following equivalents:
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| (x, , ... , , ) = (x) |
| (x, , ... , , ) = (x) |
o---------------------------------------o
o---------------------------------------o
</pre>
|-
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| (x_1, ..., x_k, ()) = x_1 · ... · x_k |
| (x_1, ..., x_k, ()) = x_1 · ... · x_k |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
Laying out the arrows on the augmented venn diagram,
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
one gets a picture of a "differential vector field".
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| |
| |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
<pre>
This really just constitutes a depiction of
This really just constitutes a depiction of
the interpretations in EU = X x Y x dX x dY
the interpretations in EU = X x Y x dX x dY
