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Directory:Jon Awbrey/Papers/Differential Logic (view source)
Revision as of 00:36, 10 June 2009
, 00:36, 10 June 2009→Note 13: centering + spacing
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
A:A & A:B & A:C
A\!:\!A & A\!:\!B & A\!:\!C
\\
\\
B:A & B:B & B:C
B\!:\!A & B\!:\!B & B\!:\!C
\\
\\
C:A & C:B & C:C
C\!:\!A & C\!:\!B & C\!:\!C
\end{bmatrix}</math>
\end{bmatrix}</math>
|}
|}
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
e_{11} & e_{12} & e_{13}
e_{11} & e_{12} & e_{13}
|}
|}
So, for example, let us suppose that we have the small universe <math>\{ A, B, C \},\!</math> and the 2-adic relation <math>m = {}^{\backprime\backprime}\, \text{mover of}\, \underline{~~~~}\, {}^{\prime\prime}</math> that is represented by this matrix:
So, for example, let us suppose that we have the small universe <math>\{ A, B, C \},\!</math> and the 2-adic relation <math>m = {}^{\backprime\backprime}\, \text{mover of}\, \underline{~~~~}\, {}^{\prime\prime}</math> that is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>
<math>\begin{bmatrix}
m_{AA} (A\!:\!A) & m_{AB} (A\!:\!B) & m_{AC} (A\!:\!C)
\begin{bmatrix}
m_{AA} (A:A) & m_{AB} (A:B) & m_{AC} (A:C)
\\
\\
m_{BA} (B:A) & m_{BB} (B:B) & m_{BC} (B:C)
m_{BA} (B\!:\!A) & m_{BB} (B\!:\!B) & m_{BC} (B\!:\!C)
\\
\\
m_{CA} (C:A) & m_{CB} (C:B) & m_{CC} (C:C)
m_{CA} (C\!:\!A) & m_{CB} (C\!:\!B) & m_{CC} (C\!:\!C)
\end{bmatrix}
\end{bmatrix}</math>
</math>
|}
|}
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>\begin{array}{l}
<math>\begin{array}{l}
A ~\text{is a mover of}~ A ~\text{and}~ B,
A ~\text{is a mover of}~ A ~\text{and}~ B,
|}
|}
In sum:
In sum, <math>m\!</math> is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>
<math>\begin{bmatrix}
1 \cdot (A\!:\!A) & 1 \cdot (A\!:\!B) & 0 \cdot (A\!:\!C)
\begin{bmatrix}
1 \cdot (A:A) & 1 \cdot (A:B) & 0 \cdot (A:C)
\\
\\
0 \cdot (B:A) & 1 \cdot (B:B) & 1 \cdot (B:C)
0 \cdot (B\!:\!A) & 1 \cdot (B\!:\!B) & 1 \cdot (B\!:\!C)
\\
\\
1 \cdot (C:A) & 0 \cdot (C:B) & 1 \cdot (C:C)
1 \cdot (C\!:\!A) & 0 \cdot (C\!:\!B) & 1 \cdot (C\!:\!C)
\end{bmatrix}
\end{bmatrix}</math>
</math>
|}
|}