Changes

Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 21:10, 18 June 2008

3,684 bytes added , 21:10, 18 June 2008

→‎Detail of Calculation for the Difference Map: new table

Line 7,540: Line 7,540:

{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Detail of Calculation for <math>\operatorname{D}f = \operatorname{E}f + f</math>'''

−

|- style="background:ghostwhite; height:~~36px~~"

+

|- style="background:ghostwhite; height:60px"

|  

−

| <math>\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}~~</math> <math>~~+ f|_{\operatorname{d}x\ \operatorname{d}y}</math>

+

|

−

| <math>\operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)}~~</math> <math>~~+ f|_{\operatorname{d}x\ (\operatorname{d}y)}</math>

+

<math>\begin{array}{cr}

−

| <math>\operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y}~~</math> <math>~~+ f|_{(\operatorname{d}x)\ \operatorname{d}y}</math>

+

& \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\

−

| <math>\operatorname{E}f|_{(\operatorname{d}x) (\operatorname{d}y)}~~</math> <math>~~+ f|_{(\operatorname{d}x) (\operatorname{d}y)}</math>

+

+ & f|_{\operatorname{d}x\ \operatorname{d}y} \\

+

= & \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \\

+

\end{array}</math>

+

|

+

<math>\begin{array}{cr}

+

& \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\

+

+ & f|_{\operatorname{d}x\ (\operatorname{d}y)} \\

+

= & \operatorname{D}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\

+

\end{array}</math>

+

|

+

<math>\begin{array}{cr}

+

& \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\

+

+ & f|_{(\operatorname{d}x)\ \operatorname{d}y} \\

+

= & \operatorname{D}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\

+

\end{array}</math>

+

|

+

<math>\begin{array}{cr}

+

& \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\

+

+ & f|_{(\operatorname{d}x)(\operatorname{d}y)} \\

+

= & \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\

+

\end{array}</math>

|- style="height:40px"

| <math>f_{0}\!</math>

−

| <math>0 + 0\!</math>

+

| <math>0 + 0 = 0\!</math>

−

| <math>0 + 0\!</math>

+

| <math>0 + 0 = 0\!</math>

−

| <math>0 + 0\!</math>

+

| <math>0 + 0 = 0\!</math>

−

| <math>0 + 0\!</math>

+

| <math>0 + 0 = 0\!</math>

|-

|

{| align="center"

|-

−

| height="~~40px~~" | <math>f_{1}\!</math>

+

| height="60px" | <math>f_{1}\!</math>

|-

−

| height="~~40px~~" | <math>f_{2}\!</math>

+

| height="60px" | <math>f_{2}\!</math>

|-

−

| height="~~40px~~" | <math>f_{4}\!</math>

+

| height="60px" | <math>f_{4}\!</math>

|-

−

| height="~~40px~~" | <math>f_{8}\!</math>

+

| height="60px" | <math>f_{8}\!</math>

|}

|

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& x & y & \operatorname{d}x & \operatorname{d}y \\

+

& x & y & \operatorname{d}x & \operatorname{d}y \\

−

+ & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\

+

+ & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\

+

= & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & (y) & \operatorname{d}x & \operatorname{d}y \\

+ & (x) & y & \operatorname{d}x & \operatorname{d}y \\

+

= & (x, & y) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & y & \operatorname{d}x & \operatorname{d}y \\

+ & x & (y) & \operatorname{d}x & \operatorname{d}y \\

+

= & (x, & y) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x) & (y) & \operatorname{d}x & \operatorname{d}y \\

+

& (x) & (y) & \operatorname{d}x & \operatorname{d}y \\

−

+ & x & y & \operatorname{d}x & \operatorname{d}y \\

+

+ & x & y & \operatorname{d}x & \operatorname{d}y \\

+

= & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & (y) & \operatorname{d}x & (\operatorname{d}y) \\

+ & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & (y) & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & y & \operatorname{d}x & (\operatorname{d}y) \\

+ & (x) & y & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & y & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\

+ & x & (y) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & (y) & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & y & \operatorname{d}x & (\operatorname{d}y) \\

+ & x & y & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & y & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & y & (\operatorname{d}x) & \operatorname{d}y \\

+ & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & (x) & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\

+ & (x) & y & (\operatorname{d}x) & \operatorname{d}y \\

+

= & (x) & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & y & (\operatorname{d}x) & \operatorname{d}y \\

+ & x & (y) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & x & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & (y) & (\operatorname{d}x) & \operatorname{d}y \\

+ & x & y & (\operatorname{d}x) & \operatorname{d}y \\

+

= & x & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x) & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& (x) (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & (x) & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & (x) (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x) & y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & (x) & y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& x & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & x & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& x & y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & x & y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" | <math>f_{3}\!</math>

+

| height="60px" | <math>f_{3}\!</math>

|-

−

| height="~~40px~~" | <math>f_{12}\!</math>

+

| height="60px" | <math>f_{12}\!</math>

|}

|

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & & \operatorname{d}x & \operatorname{d}y \\

+ & (x) & & \operatorname{d}x & \operatorname{d}y \\

+

= & 1 & & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & & \operatorname{d}x & \operatorname{d}y \\

+ & x & & \operatorname{d}x & \operatorname{d}y \\

+

= & 1 & & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & & \operatorname{d}x & (\operatorname{d}y) \\

+ & (x) & & \operatorname{d}x & (\operatorname{d}y) \\

+

= & 1 & & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & & \operatorname{d}x & (\operatorname{d}y) \\

+ & x & & \operatorname{d}x & (\operatorname{d}y) \\

+

= & 1 & & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & & (\operatorname{d}x) & \operatorname{d}y \\

+ & (x) & & (\operatorname{d}x) & \operatorname{d}y \\

+

= & 0 & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & & (\operatorname{d}x) & \operatorname{d}y \\

+ & x & & (\operatorname{d}x) & \operatorname{d}y \\

+

= & 0 & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\

+ & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& x & & (\operatorname{d}x) & (\operatorname{d}y) \\

+ & x & & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" | <math>f_{6}\!</math>

+

| height="60px" | <math>f_{6}\!</math>

|-

−

| height="~~40px~~" | <math>f_{9}\!</math>

+

| height="60px" | <math>f_{9}\!</math>

|}

|

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x, & y) & \operatorname{d}x & \operatorname{d}y \\

+

& (x , y) & \operatorname{d}x & \operatorname{d}y \\

−

+ & (x, & y) & \operatorname{d}x & \operatorname{d}y \\

+

+ & (x , y) & \operatorname{d}x & \operatorname{d}y \\

+

= & 0 & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\

+

& ((x , y)) & \operatorname{d}x & \operatorname{d}y \\

−

+ & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\

+

+ & ((x , y)) & \operatorname{d}x & \operatorname{d}y \\

+

= & 0 & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& ((x, & y)) & \operatorname{d}x & (\operatorname{d}y) \\

+

& ((x , y)) & \operatorname{d}x & (\operatorname{d}y) \\

−

+ & (x, & y) & \operatorname{d}x & (\operatorname{d}y) \\

+

+ & (x , y) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & 1 & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x, & y) & \operatorname{d}x & (\operatorname{d}y) \\

+

& (x , y) & \operatorname{d}x & (\operatorname{d}y) \\

−

+ & ((x, & y)) & \operatorname{d}x & (\operatorname{d}y) \\

+

+ & ((x , y)) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & 1 & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& ((x, & y)) & (\operatorname{d}x) & \operatorname{d}y \\

+

& ((x , y)) & (\operatorname{d}x) & \operatorname{d}y \\

−

+ & (x, & y) & (\operatorname{d}x) & \operatorname{d}y \\

+

+ & (x , y) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & 1 & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x, & y) & (\operatorname{d}x) & \operatorname{d}y \\

+

& (x , y) & (\operatorname{d}x) & \operatorname{d}y \\

−

+ & ((x, & y)) & (\operatorname{d}x) & \operatorname{d}y \\

+

+ & ((x , y)) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & 1 & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x, & y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& (x , y) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & (x, & y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & (x , y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& ((x, & y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& ((x , y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & ((x, & y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & ((x , y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" | <math>f_{5}\!</math>

+

| height="60px" | <math>f_{5}\!</math>

|-

−

| height="~~40px~~" | <math>f_{10}\!</math>

+

| height="60px" | <math>f_{10}\!</math>

|}

|

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & y & \operatorname{d}x & \operatorname{d}y \\

+ & & (y) & \operatorname{d}x & \operatorname{d}y \\

+

= & & 1 & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & (y) & \operatorname{d}x & \operatorname{d}y \\

+ & & y & \operatorname{d}x & \operatorname{d}y \\

+

= & & 1 & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & (y) & \operatorname{d}x & (\operatorname{d}y) \\

+ & & (y) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & 0 & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & y & \operatorname{d}x & (\operatorname{d}y) \\

+ & & y & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & 0 & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & y & (\operatorname{d}x) & \operatorname{d}y \\

+ & & (y) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & & 1 & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & (y) & (\operatorname{d}x) & \operatorname{d}y \\

+ & & y & (\operatorname{d}x) & \operatorname{d}y \\

+

= & & 1 & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+ & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& & y & (\operatorname{d}x) & (\operatorname{d}y) \\

+ & & y & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

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{| align="center"

|-

−

| height="~~40px~~" | <math>f_{7}\!</math>

+

| height="60px" | <math>f_{7}\!</math>

|-

−

| height="~~40px~~" | <math>f_{11}\!</math>

+

| height="60px" | <math>f_{11}\!</math>

|-

−

| height="~~40px~~" | <math>f_{13}\!</math>

+

| height="60px" | <math>f_{13}\!</math>

|-

−

| height="~~40px~~" | <math>f_{14}\!</math>

+

| height="60px" | <math>f_{14}\!</math>

|}

|

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\

+ & (x & y) & \operatorname{d}x & \operatorname{d}y \\

+

= & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& ((x) & y) & \operatorname{d}x & \operatorname{d}y \\

+ & (x & (y)) & \operatorname{d}x & \operatorname{d}y \\

+

= & (x, & y) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x & (y)) & \operatorname{d}x & \operatorname{d}y \\

+ & ((x) & y) & \operatorname{d}x & \operatorname{d}y \\

+

= & (x, & y) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x & y) & \operatorname{d}x & \operatorname{d}y \\

+ & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\

+

= & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

Line 7,918: Line 7,982:

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& ((x) & y) & \operatorname{d}x & (\operatorname{d}y) \\

+ & (x & y) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & y & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\

+ & (x & (y)) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & (y) & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x & y) & \operatorname{d}x & (\operatorname{d}y) \\

+ & ((x) & y) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & y & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x & (y)) & \operatorname{d}x & (\operatorname{d}y) \\

+ & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\

+

= & & (y) & \operatorname{d}x & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

Line 7,945: Line 8,013:

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x & (y)) & (\operatorname{d}x) & \operatorname{d}y \\

+ & (x & y) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & x & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& (x & y) & (\operatorname{d}x) & \operatorname{d}y \\

+ & (x & (y)) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & x & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\

+ & ((x) & y) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & (x) & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

& ((x) & y) & (\operatorname{d}x) & \operatorname{d}y \\

+ & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\

+

= & (x) & & (\operatorname{d}x) & \operatorname{d}y \\

\end{smallmatrix}</math>

|}

Line 7,972: Line 8,044:

{| align="center"

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x & y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & (x & y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& (x & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & (x & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& ((x) & y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & ((x) & y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|-

−

| height="~~40px~~" |

+

| height="60px" |

<math>\begin{smallmatrix}

−

& ((x) & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

& ((x) (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

−

+ & ((x) & (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

+ & ((x) (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\

+

= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\

\end{smallmatrix}</math>

|}

|- style="height:40px"

| <math>f_{15}\!</math>

−

| <math>1 + 1\!</math>

+

| <math>1 + 1 = 0\!</math>

−

| <math>1 + 1\!</math>

+

| <math>1 + 1 = 0\!</math>

−

| <math>1 + 1\!</math>

+

| <math>1 + 1 = 0\!</math>

−

| <math>1 + 1\!</math>

+

| <math>1 + 1 = 0\!</math>

|}

Jon Awbrey

12,080

edits