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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 17:11, 29 May 2008
, 17:11, 29 May 2008archive old work
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{B}^n]</math>
<math>[\mathbb{B}^n]</math>
|}<br>
|}
<br>
====Differential Propositions====
====Differential Propositions====
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{D}^n]</math>
<math>[\mathbb{D}^n]</math>
|}<br>
|}
<br>
'''…'''
'''…'''
<br>
<br>
=Work Area 1=
=Archive 1=
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{B}^n]</math>
<math>[\mathbb{B}^n]</math>
|}<br>
|}
<br>
A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{D}^n]</math>
<math>[\mathbb{D}^n]</math>
|}<br>
|}
<br>
=Work Area 2=
=Archive 2=
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
| <math>f_{0000}\!</math>
| <math>f_{0000}\!</math>
| 0 0 0 0
| 0 0 0 0
| <math>(\!|~|\!)</math>
| <math>(~)\!</math>
| false
| false
| <math>0\!</math>
| <math>0\!</math>
| <math>f_{0001}\!</math>
| <math>f_{0001}\!</math>
| 0 0 0 1
| 0 0 0 1
| <math>(\!|x|\!)(\!|y|\!)</math>
| <math>(x)(y)\!</math>
| neither x nor y
| neither x nor y
| <math>\lnot x \land \lnot y</math>
| <math>\lnot x \land \lnot y\!</math>
|-
|-
| <math>f_{2}\!</math>
| <math>f_{2}\!</math>
| <math>f_{0010}\!</math>
| <math>f_{0010}\!</math>
| 0 0 1 0
| 0 0 1 0
| <math>(\!|x|\!)\ y</math>
| <math>(x)\ y\!</math>
| y and not x
| y and not x
| <math>\lnot x \land y</math>
| <math>\lnot x \land y\!</math>
|-
|-
| <math>f_{3}\!</math>
| <math>f_{3}\!</math>
| <math>f_{0011}\!</math>
| <math>f_{0011}\!</math>
| 0 0 1 1
| 0 0 1 1
| <math>(\!|x|\!)</math>
| <math>(x)\!</math>
| not x
| not x
| <math>\lnot x</math>
| <math>\lnot x\!</math>
|-
|-
| <math>f_{4}\!</math>
| <math>f_{4}\!</math>
| <math>f_{0100}\!</math>
| <math>f_{0100}\!</math>
| 0 1 0 0
| 0 1 0 0
| <math>x\ (\!|y|\!)</math>
| <math>x\ (y)\!</math>
| x and not y
| x and not y
| <math>x \land \lnot y</math>
| <math>x \land \lnot y\!</math>
|-
|-
| <math>f_{5}\!</math>
| <math>f_{5}\!</math>
| <math>f_{0101}\!</math>
| <math>f_{0101}\!</math>
| 0 1 0 1
| 0 1 0 1
| <math>(\!|y|\!)</math>
| <math>(y)\!</math>
| not y
| not y
| <math>\lnot y</math>
| <math>\lnot y\!</math>
|-
|-
| <math>f_{6}\!</math>
| <math>f_{6}\!</math>
| <math>f_{0110}\!</math>
| <math>f_{0110}\!</math>
| 0 1 1 0
| 0 1 1 0
| <math>(\!|x,\ y|\!)</math>
| <math>(x,\ y)\!</math>
| x not equal to y
| x not equal to y
| <math>x \ne y</math>
| <math>x \ne y\!</math>
|-
|-
| <math>f_{7}\!</math>
| <math>f_{7}\!</math>
| <math>f_{0111}\!</math>
| <math>f_{0111}\!</math>
| 0 1 1 1
| 0 1 1 1
| <math>(\!|x\ y|\!)</math>
| <math>(x\ y)\!</math>
| not both x and y
| not both x and y
| <math>\lnot x \lor \lnot y</math>
| <math>\lnot x \lor \lnot y\!</math>
|-
|-
| <math>f_{8}\!</math>
| <math>f_{8}\!</math>
| <math>f_{1000}\!</math>
| <math>f_{1000}\!</math>
| 1 0 0 0
| 1 0 0 0
| <math>x\ y</math>
| <math>x\ y\!</math>
| x and y
| x and y
| <math>x \land y</math>
| <math>x \land y\!</math>
|-
|-
| <math>f_{9}\!</math>
| <math>f_{9}\!</math>
| <math>f_{1001}\!</math>
| <math>f_{1001}\!</math>
| 1 0 0 1
| 1 0 0 1
| <math>(\!|(\!|x,\ y|\!)|\!)</math>
| <math>((x,\ y))\!</math>
| x equal to y
| x equal to y
| <math>x = y\!</math>
| <math>x = y\!</math>
| <math>f_{1011}\!</math>
| <math>f_{1011}\!</math>
| 1 0 1 1
| 1 0 1 1
| <math>(\!|x\ (\!|y|\!)|\!)</math>
| <math>(x\ (y))\!</math>
| not x without y
| not x without y
| <math>x \Rightarrow y</math>
| <math>x \Rightarrow y\!</math>
|-
|-
| <math>f_{12}\!</math>
| <math>f_{12}\!</math>
| <math>f_{1101}\!</math>
| <math>f_{1101}\!</math>
| 1 1 0 1
| 1 1 0 1
| <math>(\!|(\!|x|\!)\ y|\!)</math>
| <math>((x)\ y)\!</math>
| not y without x
| not y without x
| <math>x \Leftarrow y</math>
| <math>x \Leftarrow y\!</math>
|-
|-
| <math>f_{14}\!</math>
| <math>f_{14}\!</math>
| <math>f_{1110}\!</math>
| <math>f_{1110}\!</math>
| 1 1 1 0
| 1 1 1 0
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| <math>((x)(y))\!</math>
| x or y
| x or y
| <math>x \lor y</math>
| <math>x \lor y\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>f_{1111}\!</math>
| 1 1 1 1
| 1 1 1 1
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>((~))\!</math>
| true
| true
| <math>1\!</math>
| <math>1\!</math>
|}<br>
|}
<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 2. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" |
| style="width:16%" | <math>\mathcal{L}_1</math>
| style="width:16%" | <math>f\!</math>
| style="width:16%" | <math>\mathcal{L}_2</math>
| style="width:16%" | <math>\operatorname{E}f|_{xy}</math>
| style="width:16%" | <math>\mathcal{L}_3</math>
| style="width:16%" | <math>\operatorname{E}f|_{x(\!|y|\!)}</math>
| style="width:16%" | <math>\mathcal{L}_4</math>
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)y}</math>
| style="width:16%" | <math>\mathcal{L}_5</math>
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
| style="width:16%" | <math>\mathcal{L}_6</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\!</math> :
| 1 1 0 0
|
|
|
|- style="background:ghostwhite"
|
| align="right" | <math>y\!</math> :
| 1 0 1 0
|
|
|
|-
|-
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
| <math>f_{0000}\!</math>
| 0 0 0 0
| <math>(\!|~|\!)</math>
| <math>(\!|~|\!)</math>
| <math>(\!|~|\!)</math>
| false
| <math>0\!</math>
| <math>(\!|~|\!)</math>
|-
|-
| <math>f_{1}\!</math>
| <math>f_{1}\!</math>
| <math>f_{0001}\!</math>
| 0 0 0 1
| <math>(\!|x|\!)(\!|y|\!)</math>
| <math>(\!|x|\!)(\!|y|\!)</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| neither x nor y
| <math>\lnot x \land \lnot y</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
|-
|-
| <math>f_{2}\!</math>
| <math>f_{2}\!</math>
| <math>f_{0010}\!</math>
| 0 0 1 0
| <math>(\!|x|\!)\ y</math>
| y and not x
| <math>\lnot x \land y</math>
| <math>f_{4}\!</math>
| <math>x (\!|y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
|-
|-
| <math>f_{3}\!</math>
| <math>f_{3}\!</math>
| <math>f_{0011}\!</math>
| 0 0 1 1
| <math>(\!|x|\!)</math>
| <math>(\!|x|\!)</math>
| <math>\operatorname{d}x</math>
| not x
| <math>\operatorname{d}x</math>
| <math>\lnot x</math>
| <math>(\!|\operatorname{d}x|\!)</math>
|-
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>f_{4}\!</math>
| <math>f_{0100}\!</math>
| 0 1 0 0
| <math>x\ (\!|y|\!)</math>
| x and not y
| <math>x \land \lnot y</math>
|-
|-
| <math>f_{12}\!</math>
| <math>f_{5}\!</math>
| <math>x\!</math>
| <math>f_{0101}\!</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| 0 1 0 1
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>(\!|y|\!)</math>
| <math>\operatorname{d}x</math>
| not y
| <math>\operatorname{d}x</math>
| <math>\lnot y</math>
|-
|-
| <math>f_{6}\!</math>
| <math>f_{6}\!</math>
| <math>(\!|x, y|\!)</math>
| <math>f_{0110}\!</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| 0 1 1 0
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>(\!|x,\ y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| x not equal to y
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>x \ne y</math>
|-
| <math>f_{7}\!</math>
| <math>f_{0111}\!</math>
| 0 1 1 1
| <math>(\!|x\ y|\!)</math>
| not both x and y
| <math>\lnot x \lor \lnot y</math>
|-
|-
| <math>f_{9}\!</math>
| <math>f_{8}\!</math>
| <math>(\!|(\!|x, y|\!)|\!)</math>
| <math>f_{1000}\!</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| 1 0 0 0
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>x\ y</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| x and y
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>x \land y</math>
|-
|-
| <math>f_{5}\!</math>
| <math>f_{9}\!</math>
| <math>(\!|y|\!)</math>
| <math>f_{1001}\!</math>
| <math>\operatorname{d}y</math>
| 1 0 0 1
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>(\!|(\!|x,\ y|\!)|\!)</math>
| <math>\operatorname{d}y</math>
| x equal to y
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>x = y\!</math>
|-
|-
| <math>f_{10}\!</math>
| <math>f_{10}\!</math>
| <math>f_{1010}\!</math>
| 1 0 1 0
| <math>y\!</math>
| y
| <math>y\!</math>
| <math>y\!</math>
|-
|-
| <math>f_{11}\!</math>
| <math>f_{11}\!</math>
| <math>(\!|x (\!|y|\!)|\!)</math>
| <math>f_{1011}\!</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| 1 0 1 1
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|x\ (\!|y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| not x without y
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>x \Rightarrow y</math>
|-
| <math>f_{12}\!</math>
| <math>f_{1100}\!</math>
| 1 1 0 0
| <math>x\!</math>
| x
| <math>x\!</math>
|-
|-
| <math>f_{13}\!</math>
| <math>f_{13}\!</math>
| <math>(\!|(\!|x|\!) y|\!)</math>
| <math>f_{1101}\!</math>
| 1 1 0 1
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>(\!|(\!|x|\!)\ y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| not y without x
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>x \Leftarrow y</math>
|-
|-
| <math>f_{14}\!</math>
| <math>f_{14}\!</math>
| <math>f_{1110}\!</math>
| 1 1 1 0
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| x or y
| <math>x \lor y</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| 1 1 1 1
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>(\!|(\!|~|\!)|\!)</math>
| true
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>1\!</math>
| <math>(\!|(\!|~|\!)|\!)</math>
|}
<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 2. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" | <math>\mathcal{L}_1</math>
| style="width:16%" |
| style="width:16%" | <math>\mathcal{L}_2</math>
| style="width:16%" | <math>f\!</math>
| style="width:16%" | <math>\mathcal{L}_3</math>
| style="width:16%" | <math>\operatorname{E}f|_{xy}</math>
| style="width:16%" | <math>\mathcal{L}_4</math>
| style="width:16%" | <math>\operatorname{E}f|_{x(\!|y|\!)}</math>
| style="width:16%" | <math>\mathcal{L}_5</math>
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)y}</math>
| style="width:16%" | <math>\mathcal{L}_6</math>
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\!</math> :
| 1 1 0 0
|
|-
|-
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
| <math>f_{0000}\!</math>
| <math>(\!|~|\!)</math>
| 0 0 0 0
| <math>(\!|~|\!)</math>
| <math>(~)\!</math>
| <math>(\!|~|\!)</math>
| false
| <math>(\!|~|\!)</math>
| <math>0\!</math>
| <math>(\!|~|\!)</math>
|-
|-
| <math>f_{1}\!</math>
| <math>f_{1}\!</math>
| <math>f_{0001}\!</math>
| <math>(\!|x|\!)(\!|y|\!)</math>
| 0 0 0 1
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>(x)(y)\!</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| neither x nor y
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>\lnot x \land \lnot y\!</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
|-
|-
| <math>f_{2}\!</math>
| <math>f_{2}\!</math>
| <math>f_{0010}\!</math>
| <math>(\!|x|\!) y</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| <math>(x)\ y\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>\lnot x \land y\!</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>f_{3}\!</math>
| <math>f_{0011}\!</math>
| <math>(x)\!</math>
| not x
| <math>\lnot x\!</math>
|-
|-
| <math>f_{4}\!</math>
| <math>f_{4}\!</math>
| <math>f_{0100}\!</math>
| <math>x (\!|y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>x\ (y)\!</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| x and not y
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>x \land \lnot y\!</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
|-
|-
| <math>f_{5}\!</math>
| <math>f_{8}\!</math>
| <math>f_{0101}\!</math>
| <math>x y\!</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>(y)\!</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| not y
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| <math>\lnot y\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
|-
|-
| <math>f_{6}\!</math>
| <math>f_{3}\!</math>
| <math>f_{0110}\!</math>
| <math>(\!|x|\!)</math>
| 0 1 1 0
| <math>\operatorname{d}x</math>
| <math>(x,\ y)\!</math>
| <math>\operatorname{d}x</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>x \ne y\!</math>
| <math>(\!|\operatorname{d}x|\!)</math>
|-
|-
| <math>f_{7}\!</math>
| <math>f_{12}\!</math>
| <math>f_{0111}\!</math>
| <math>x\!</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>(x\ y)\!</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| not both x and y
| <math>\operatorname{d}x</math>
| <math>\lnot x \lor \lnot y\!</math>
| <math>\operatorname{d}x</math>
|-
|-
| <math>f_{8}\!</math>
| <math>f_{6}\!</math>
| <math>f_{1000}\!</math>
| <math>(\!|x, y|\!)</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>x\ y\!</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| x and y
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>x \land y\!</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
|-
|-
| <math>f_{9}\!</math>
| <math>f_{9}\!</math>
| <math>f_{1001}\!</math>
| <math>(\!|(\!|x, y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>((x,\ y))\!</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| x equal to y
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>x = y\!</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
|-
|-
| <math>f_{10}\!</math>
| <math>f_{5}\!</math>
| <math>f_{1010}\!</math>
| <math>(\!|y|\!)</math>
| 1 0 1 0
| <math>\operatorname{d}y</math>
| <math>y\!</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| y
| <math>\operatorname{d}y</math>
| <math>(\!|\operatorname{d}y|\!)</math>
|-
| <math>f_{10}\!</math>
| <math>y\!</math>
| <math>y\!</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>\operatorname{d}y</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>\operatorname{d}y</math>
|-
| <math>f_{7}\!</math>
| <math>(\!|x y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
|-
|-
| <math>f_{11}\!</math>
| <math>f_{11}\!</math>
| <math>f_{1011}\!</math>
| <math>(\!|x (\!|y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>(x\ (y))\!</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| not x without y
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>x \Rightarrow y\!</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>f_{12}\!</math>
| <math>f_{1100}\!</math>
| <math>x\!</math>
| x
| <math>x\!</math>
|-
|-
| <math>f_{13}\!</math>
| <math>f_{13}\!</math>
| <math>f_{1101}\!</math>
| <math>(\!|(\!|x|\!) y|\!)</math>
| 1 1 0 1
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>((x)\ y)\!</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>x \Leftarrow y\!</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
|-
|-
| <math>f_{14}\!</math>
| <math>f_{14}\!</math>
| <math>f_{1110}\!</math>
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>((x)(y))\!</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| x or y
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>x \lor y\!</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>(\!|(\!|~|\!)|\!)</math>
| 1 1 1 1
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>((~))\!</math>
| <math>(\!|(\!|~|\!)|\!)</math>
| true
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>1\!</math>
| <math>(\!|(\!|~|\!)|\!)</math>
|}
|}
<br>
<br>
===Variant 2===
=Work Area 1=
==Propositional Forms on Two Variables==
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> For future reference, I will set here a few Tables that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
===Variant 1===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
<br>
<br>
===Variant 3===
===Variant 2===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
<br>
<br>
===Variant 4===
===Variant 3===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
<br>
<br>
===Variant 5===
===Variant 4===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
|}<br>
|}
<br>
<pre>
<pre>