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Revision as of 06:12, 31 May 2008
, 06:12, 31 May 2008archive old work
{| 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 1. Propositional Forms on Two Variables'''
|+ '''Propositional Forms on Two Variables'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" | <math>\mathcal{L}_1</math>
| style="width:16%" | <math>\mathcal{L}_1</math>
|}
|}
<br>
<br>
=Archive 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%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|+ '''Propositional Forms on Two Variables'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" | <math>\mathcal{L}_1</math>
| style="width:16%" | <math>\mathcal{L}_1</math>
{| 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>'''
|+ '''<math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" |
| style="width:16%" |
<br>
<br>
=Archive 3=
=Archive 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%"
<br>
<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 3. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:ghostwhite; height:36px"
|- style="background:ghostwhite; height:36px"
|
|
|
| <math>f\!</math>
{| align="right" style="background:ghostwhite; text-align:right"
| <math>\operatorname{E}f|_{xy}</math>
| <math>\operatorname{E}f|_{x(y)}</math>
| <math>\operatorname{E}f|_{(x)y}</math>
| <math>\operatorname{E}f|_{(x)(y)}</math>
|-
|-
| <math>x\!</math> :
| <math>f_{0}\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
|-
|-
| <math>y\!</math> :
| <math>f_{1}\!</math>
|}
| <math>(x)(y)\!</math>
|
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
{| align="center" style="background:ghostwhite"
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
|-
|-
| 1 1 0 0
| <math>f_{2}\!</math>
|-
| <math>(x) y\!</math>
| 1 0 1 0
| <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_{4}\!</math>
{| align="center"
| <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>
|-
|-
| height="36px" | <p><math>f_{0}\!</math></p>
| <math>f_{8}\!</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>
|-
|-
| height="36px" | <p><math>f_{1}\!</math></p>
| <math>f_{3}\!</math>
|-
| <math>(x)\!</math>
| height="36px" | <p><math>f_{2}\!</math></p>
| <math>\operatorname{d}x\!</math>
| <math>\operatorname{d}x\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\operatorname{d}x)\!</math>
|-
|-
| height="36px" | <p><math>f_{3}\!</math></p>
| <math>f_{12}\!</math>
| <math>x\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\operatorname{d}x\!</math>
|-
|-
| height="36px" | <p><math>f_{4}\!</math></p>
| <math>f_{6}\!</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>
|-
|-
| height="36px" | <p><math>f_{5}\!</math></p>
| <math>f_{9}\!</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>
|-
|-
| height="36px" | <p><math>f_{6}\!</math></p>
| <math>f_{5}\!</math>
| <math>(y)\!</math>
| <math>\operatorname{d}y\!</math>
|}
| <math>(\operatorname{d}y)\!</math>
|
| <math>\operatorname{d}y\!</math>
{| align="center"
| <math>(\operatorname{d}y)\!</math>
|-
|-
| height="36px" | <p><math>f_{0000}\!</math></p>
| <math>f_{10}\!</math>
| <math>y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
|-
|-
| height="36px" | <p><math>f_{0001}\!</math></p>
| <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>
|-
|-
| height="36px" | <p><math>f_{0010}\!</math></p>
| <math>f_{11}\!</math>
|-
| <math>(x (y))\!</math>
| height="36px" | <p><math>f_{0011}\!</math></p>
| <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>
|-
|-
| height="36px" | <p><math>f_{0100}\!</math></p>
| <math>f_{13}\!</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>
|-
|-
| height="36px" | <p><math>f_{0101}\!</math></p>
| <math>f_{14}\!</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>
|-
|-
| height="36px" | <p><math>f_{0110}\!</math></p>
| <math>f_{15}\!</math>
|-
| <math>((~))\!</math>
| height="36px" | <p><math>f_{0111}\!</math></p>
| <math>((~))\!</math>
|}
| <math>((~))\!</math>
|
| <math>((~))\!</math>
{| align="center"
| <math>((~))\!</math>
|}<br>
=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.
===Table 1===
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:ghostwhite; height:36px"
| <math>\mathcal{L}_1</math>
| <math>\mathcal{L}_2</math>
| <math>\mathcal{L}_3</math>
| <math>\mathcal{L}_4</math>
| <math>\mathcal{L}_5</math>
| <math>\mathcal{L}_6</math>
|- style="background:ghostwhite; height:48px"
|
|
{| align="right" style="background:ghostwhite; text-align:right"
|-
|-
| height="36px" | 0 0 0 0
| <math>x\!</math> :
|-
|-
| height="36px" | 0 0 0 1
| <math>y\!</math> :
|}
|
{| align="center" style="background:ghostwhite"
|-
|-
| height="36px" | 0 0 1 0
| 1 1 0 0
|-
|-
| height="36px" | 0 0 1 1
| 1 0 1 0
|-
|}
| height="36px" | 0 1 0 0
|
|-
|
| height="36px" | 0 1 0 1
|
|-
|-
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(~)\!</math></p>
| height="36px" | <p><math>f_{0}\!</math></p>
|-
|-
| height="36px" | <p><math>(x)(y)\!</math></p>
| height="36px" | <p><math>f_{1}\!</math></p>
|-
|-
| height="36px" | <p><math>(x)\ y\!</math></p>
| height="36px" | <p><math>f_{2}\!</math></p>
|-
|-
| height="36px" | <p><math>(x)\!</math></p>
| height="36px" | <p><math>f_{3}\!</math></p>
|-
|-
| height="36px" | <p><math>x\ (y)\!</math></p>
| height="36px" | <p><math>f_{4}\!</math></p>
|-
|-
| height="36px" | <p><math>(y)\!</math></p>
| height="36px" | <p><math>f_{5}\!</math></p>
|-
|-
| height="36px" | <p><math>(x,\ y)\!</math></p>
| height="36px" | <p><math>f_{6}\!</math></p>
|-
|-
| height="36px" | <p><math>(x\ y)\!</math></p>
| height="36px" | <p><math>f_{7}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{false}</math></p>
| height="36px" | <p><math>f_{0000}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
| height="36px" | <p><math>f_{0001}\!</math></p>
|-
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>f_{0010}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
| height="36px" | <p><math>f_{0011}\!</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>f_{0100}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
| height="36px" | <p><math>f_{0101}\!</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
| height="36px" | <p><math>f_{0110}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>f_{0111}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>0\!</math></p>
| height="36px" | 0 0 0 0
|-
|-
| height="36px" | <p><math>\lnot x \land \lnot y</math></p>
| height="36px" | 0 0 0 1
|-
|-
| height="36px" | <p><math>\lnot x \land y</math></p>
| height="36px" | 0 0 1 0
|-
|-
| height="36px" | <p><math>\lnot x</math></p>
| height="36px" | 0 0 1 1
|-
|-
| height="36px" | <p><math>x \land \lnot y</math></p>
| height="36px" | 0 1 0 0
|-
|-
| height="36px" | <p><math>\lnot y</math></p>
| height="36px" | 0 1 0 1
|-
|-
| height="36px" | <p><math>x \ne y</math></p>
| height="36px" | 0 1 1 0
|-
|-
| height="36px" | <p><math>\lnot x \lor \lnot y</math></p>
| height="36px" | 0 1 1 1
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{8}\!</math></p>
| height="36px" | <p><math>(~)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{9}\!</math></p>
| height="36px" | <p><math>(x)(y)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{10}\!</math></p>
| height="36px" | <p><math>(x)\ y\!</math></p>
|-
|-
| height="36px" | <p><math>f_{11}\!</math></p>
| height="36px" | <p><math>(x)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{12}\!</math></p>
| height="36px" | <p><math>x\ (y)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{13}\!</math></p>
| height="36px" | <p><math>(y)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{14}\!</math></p>
| height="36px" | <p><math>(x,\ y)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{15}\!</math></p>
| height="36px" | <p><math>(x\ y)\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{1000}\!</math></p>
| height="36px" | <p><math>\operatorname{false}</math></p>
|-
|-
| height="36px" | <p><math>f_{1001}\!</math></p>
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{1010}\!</math></p>
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
|-
|-
| height="36px" | <p><math>f_{1011}\!</math></p>
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
|-
|-
| height="36px" | <p><math>f_{1100}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{1101}\!</math></p>
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{1110}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{1111}\!</math></p>
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | 1 0 0 0
| height="36px" | <p><math>0\!</math></p>
|-
|-
| height="36px" | 1 0 0 1
| height="36px" | <p><math>\lnot x \land \lnot y</math></p>
|-
|-
| height="36px" | 1 0 1 0
| height="36px" | <p><math>\lnot x \land y</math></p>
|-
|-
| height="36px" | 1 0 1 1
| height="36px" | <p><math>\lnot x</math></p>
|-
|-
| height="36px" | 1 1 0 0
| height="36px" | <p><math>x \land \lnot y</math></p>
|-
|-
| height="36px" | 1 1 0 1
| height="36px" | <p><math>\lnot y</math></p>
|-
|-
| height="36px" | 1 1 1 0
| height="36px" | <p><math>x \ne y</math></p>
|-
|-
| height="36px" | 1 1 1 1
| height="36px" | <p><math>\lnot x \lor \lnot y</math></p>
|}
|}
|-
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ y\!</math></p>
| height="36px" | <p><math>f_{8}\!</math></p>
|-
|-
| height="36px" | <p><math>((x,\ y))\!</math></p>
| height="36px" | <p><math>f_{9}\!</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>f_{10}\!</math></p>
|-
|-
| height="36px" | <p><math>(x\ (y))\!</math></p>
| height="36px" | <p><math>f_{11}\!</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>f_{12}\!</math></p>
|-
|-
| height="36px" | <p><math>((x)\ y)\!</math></p>
| height="36px" | <p><math>f_{13}\!</math></p>
|-
|-
| height="36px" | <p><math>((x)(y))\!</math></p>
| height="36px" | <p><math>f_{14}\!</math></p>
|-
|-
| height="36px" | <p><math>((~))\!</math></p>
| height="36px" | <p><math>f_{15}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>f_{1000}\!</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
| height="36px" | <p><math>f_{1001}\!</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>f_{1010}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>f_{1011}\!</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>f_{1100}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>f_{1101}\!</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
| height="36px" | <p><math>f_{1110}\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{true}</math></p>
| height="36px" | <p><math>f_{1111}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x \land y</math></p>
| height="36px" | 1 0 0 0
|-
|-
| height="36px" | <p><math>x = y\!</math></p>
| height="36px" | 1 0 0 1
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | 1 0 1 0
|-
|-
| height="36px" | <p><math>x \Rightarrow y</math></p>
| height="36px" | 1 0 1 1
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | 1 1 0 0
|-
|-
| height="36px" | <p><math>x \Leftarrow y</math></p>
| height="36px" | 1 1 0 1
|-
|-
| height="36px" | <p><math>x \lor y</math></p>
| height="36px" | 1 1 1 0
|-
|-
| height="36px" | <p><math>1\!</math></p>
| height="36px" | 1 1 1 1
|}
|}
|}
|
{| align="center"
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|-
|-
| <math>x\!</math> :
| height="36px" | <p><math>x\ y\!</math></p>
|-
|-
| <math>y\!</math> :
| height="36px" | <p><math>((x,\ y))\!</math></p>
|-
|-
| 1 1 0 0
| height="36px" | <p><math>y\!</math></p>
|-
|-
| 1 0 1 0
| height="36px" | <p><math>(x\ (y))\!</math></p>
| <p><math>f_{0}\!</math></p>
|-
|-
|
| height="36px" | <p><math>x\!</math></p>
|-
|-
| height="36px" | <p><math>f_{1}\!</math></p>
| height="36px" | <p><math>((x)\ y)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{2}\!</math></p>
| height="36px" | <p><math>((x)(y))\!</math></p>
|-
|-
| height="36px" | <p><math>f_{4}\!</math></p>
| height="36px" | <p><math>((~))\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{0001}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{0010}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{0100}\!</math></p>
| height="36px" | <p><math>y\!</math></p>
|-
|-
| height="36px" | <p><math>f_{1000}\!</math></p>
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
|-
|-
| height="36px" | <p>0 0 0 1</p>
| height="36px" | <p><math>x\!</math></p>
|-
|-
| height="36px" | <p>0 0 1 0</p>
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
|-
|-
| height="36px" | <p>0 1 0 0</p>
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
|-
|-
| height="36px" | <p>1 0 0 0</p>
| height="36px" | <p><math>\operatorname{true}</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(x)(y)\!</math></p>
| height="36px" | <p><math>x \land y</math></p>
|-
|-
| height="36px" | <p><math>(x)\ y\!</math></p>
| height="36px" | <p><math>x = y\!</math></p>
|-
|-
| height="36px" | <p><math>x\ (y)\!</math></p>
| height="36px" | <p><math>y\!</math></p>
|-
|-
| height="36px" | <p><math>x\ y\!</math></p>
| height="36px" | <p><math>x \Rightarrow y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
| height="36px" | <p><math>x\!</math></p>
|-
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>x \Leftarrow y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>x \lor y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>1\!</math></p>
|}
|}
|}
<br>
===Table 2===
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 2. Propositional Forms on Two Variables'''
|- style="background:ghostwhite; height:36px"
| <math>\mathcal{L}_1</math>
| <math>\mathcal{L}_2</math>
| <math>\mathcal{L}_3</math>
| <math>\mathcal{L}_4</math>
| <math>\mathcal{L}_5</math>
| <math>\mathcal{L}_6</math>
|- style="background:ghostwhite; height:48px"
|
|
|
{| align="center"
{| align="right" style="background:ghostwhite; text-align:right"
|-
|-
| <math>x\!</math> :
|-
|-
| <math>y\!</math> :
|}
|
{| align="center" style="background:ghostwhite"
|-
|-
| height="36px" | <p><math>x \land \lnot y</math></p>
| 1 1 0 0
|-
|-
| height="36px" | <p><math>x \land y</math></p>
| 1 0 1 0
|}
|}
|
|
|
|- style="height:36px"
| <p><math>f_{0}\!</math></p>
| <p><math>f_{0000}\!</math></p>
| <p>0 0 0 0</p>
| <p><math>(~)\!</math></p>
| <p><math>\operatorname{false}</math></p>
| <p><math>1\!</math></p>
|-
|-
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{3}\!</math></p>
| height="36px" | <p><math>f_{1}\!</math></p>
|-
|-
| height="36px" | <p><math>f_{12}\!</math></p>
| height="36px" | <p><math>f_{2}\!</math></p>
|-
|-
| height="36px" | <p><math>f_{0011}\!</math></p>
| height="36px" | <p><math>f_{4}\!</math></p>
|-
|-
| height="36px" | <p><math>f_{1100}\!</math></p>
| height="36px" | <p><math>f_{8}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p>0 0 1 1</p>
| height="36px" | <p><math>f_{0001}\!</math></p>
|-
| height="36px" | <p><math>f_{0010}\!</math></p>
|-
| height="36px" | <p><math>f_{0100}\!</math></p>
|-
|-
| height="36px" | <p>1 1 0 0</p>
| height="36px" | <p><math>f_{1000}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(x)\!</math></p>
| height="36px" | <p>0 0 0 1</p>
|-
| height="36px" | <p>0 0 1 0</p>
|-
| height="36px" | <p>0 1 0 0</p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p>1 0 0 0</p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
| height="36px" | <p><math>(x)(y)\!</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>(x)\ y\!</math></p>
|-
|-
| height="36px" | <p><math>\lnot x</math></p>
| height="36px" | <p><math>x\ (y)\!</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>x\ y\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{6}\!</math></p>
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{9}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{0110}\!</math></p>
| height="36px" | <p><math>\lnot x \land \lnot y</math></p>
|-
|-
| height="36px" | <p><math>f_{1001}\!</math></p>
| height="36px" | <p><math>\lnot x \land y</math></p>
|-
|-
| height="36px" | <p>0 1 1 0</p>
| height="36px" | <p><math>x \land \lnot y</math></p>
|-
|-
| height="36px" | <p>1 0 0 1</p>
| height="36px" | <p><math>x \land y</math></p>
|}
|}
|-
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(x,\ y)\!</math></p>
| height="36px" | <p><math>f_{3}\!</math></p>
|-
|-
| height="36px" | <p><math>((x,\ y))\!</math></p>
| height="36px" | <p><math>f_{12}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
| height="36px" | <p><math>f_{0011}\!</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
| height="36px" | <p><math>f_{1100}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x \ne y</math></p>
| height="36px" | <p>0 0 1 1</p>
|-
|-
| height="36px" | <p><math>x = y\!</math></p>
| height="36px" | <p>1 1 0 0</p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{5}\!</math></p>
| height="36px" | <p><math>(x)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{10}\!</math></p>
| height="36px" | <p><math>x\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{0101}\!</math></p>
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
|-
|-
| height="36px" | <p><math>f_{1010}\!</math></p>
| height="36px" | <p><math>x\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p>0 1 0 1</p>
| height="36px" | <p><math>\lnot x</math></p>
|-
|-
| height="36px" | <p>1 0 1 0</p>
| height="36px" | <p><math>x\!</math></p>
|}
|}
|-
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(y)\!</math></p>
| height="36px" | <p><math>f_{6}\!</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>f_{9}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
| height="36px" | <p><math>f_{0110}\!</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>f_{1001}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\lnot y</math></p>
| height="36px" | <p>0 1 1 0</p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p>1 0 0 1</p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{7}\!</math></p>
| height="36px" | <p><math>(x,\ y)\!</math></p>
|-
|-
| height="36px" | <p><math>f_{11}\!</math></p>
| height="36px" | <p><math>((x,\ y))\!</math></p>
|}
|
{| align="center"
|-
|-
| height="36px" | <p><math>f_{13}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
|-
|-
| height="36px" | <p><math>f_{14}\!</math></p>
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{0111}\!</math></p>
| height="36px" | <p><math>x \ne y</math></p>
|-
| height="36px" | <p><math>x = y\!</math></p>
|}
|-
|-
| height="36px" | <p><math>f_{1011}\!</math></p>
|
{| align="center"
|-
|-
| height="36px" | <p><math>f_{1101}\!</math></p>
| height="36px" | <p><math>f_{5}\!</math></p>
|-
|-
| height="36px" | <p><math>f_{1110}\!</math></p>
| height="36px" | <p><math>f_{10}\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p>0 1 1 1</p>
| height="36px" | <p><math>f_{0101}\!</math></p>
|-
|-
| height="36px" | <p>1 0 1 1</p>
| height="36px" | <p><math>f_{1010}\!</math></p>
|}
|
{| align="center"
|-
|-
| height="36px" | <p>1 1 0 1</p>
| height="36px" | <p>0 1 0 1</p>
|-
|-
| height="36px" | <p>1 1 1 0</p>
| height="36px" | <p>1 0 1 0</p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(x\ y)\!</math></p>
| height="36px" | <p><math>(y)\!</math></p>
|-
|-
| height="36px" | <p><math>(x\ (y))\!</math></p>
| height="36px" | <p><math>y\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>y\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\lnot x \lor \lnot y</math></p>
| height="36px" | <p><math>\lnot y</math></p>
|-
|-
| height="36px" | <p><math>x \Rightarrow y</math></p>
| height="36px" | <p><math>y\!</math></p>
|}
|-
|
{| align="center"
|-
| height="36px" | <p><math>f_{7}\!</math></p>
|-
| height="36px" | <p><math>f_{11}\!</math></p>
|-
|-
| height="36px" | <p><math>x \Leftarrow y</math></p>
| height="36px" | <p><math>f_{13}\!</math></p>
|-
|-
| height="36px" | <p><math>x \lor y</math></p>
| height="36px" | <p><math>f_{14}\!</math></p>
|}
|}
|- style="height:36px"
|
| <p><math>f_{15}\!</math></p>
{| align="center"
| <p><math>f_{1111}\!</math></p>
|-
| <p>1 1 1 1</p>
| height="36px" | <p><math>f_{0111}\!</math></p>
| <p><math>((~))\!</math></p>
|-
| <p><math>\operatorname{true}</math></p>
| height="36px" | <p><math>f_{1011}\!</math></p>
|-
| height="36px" | <p><math>f_{1101}\!</math></p>
|-
| height="36px" | <p><math>f_{1110}\!</math></p>
|}
|}
|
{| align="center"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|-
|-
| <math>f_{0}\!</math>
| height="36px" | <p>0 1 1 1</p>
|-
|-
| <math>f_{1}\!</math>
| height="36px" | <p>1 0 1 1</p>
| <math>(x)(y)\!</math>
|-
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| height="36px" | <p>1 1 0 1</p>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
|-
|-
| <math>f_{2}\!</math>
| height="36px" | <p>1 1 1 0</p>
| <math>(x) y\!</math>
|}
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
|
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
{| align="center"
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
|-
|-
| <math>f_{4}\!</math>
| height="36px" | <p><math>(x\ y)\!</math></p>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
|-
|-
| <math>f_{8}\!</math>
| height="36px" | <p><math>(x\ (y))\!</math></p>
| <math>x y\!</math>
|-
|-
| <math>f_{3}\!</math>
| height="36px" | <p><math>((x)\ y)\!</math></p>
|-
|-
| <math>f_{12}\!</math>
| height="36px" | <p><math>((x)(y))\!</math></p>
| <math>x\!</math>
|}
|
{| align="center"
| <math>\operatorname{d}x\!</math>
| <math>\operatorname{d}x\!</math>
|-
|-
| <math>f_{6}\!</math>
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| <math>(x, y)\!</math>
|-
|-
| <math>f_{9}\!</math>
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| <math>((x, y))\!</math>
|-
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
|-
|-
| <math>f_{5}\!</math>
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
|}
|
{| align="center"
| <math>\operatorname{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
|-
|-
| <math>f_{10}\!</math>
| height="36px" | <p><math>\lnot x \lor \lnot y</math></p>
| <math>y\!</math>
|-
|-
| <math>f_{7}\!</math>
| height="36px" | <p><math>x \Rightarrow y</math></p>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
|-
|-
| <math>f_{11}\!</math>
| height="36px" | <p><math>x \Leftarrow y</math></p>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
|-
|-
| <math>f_{13}\!</math>
| height="36px" | <p><math>x \lor y</math></p>
| <math>((x) y)\!</math>
|}
|- style="height:36px"
| <p><math>f_{15}\!</math></p>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <p><math>f_{1111}\!</math></p>
| <p>1 1 1 1</p>
|-
| <p><math>((~))\!</math></p>
| <math>f_{14}\!</math>
| <p><math>\operatorname{true}</math></p>
| <p><math>1\!</math></p>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
|}
<br>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
===Table 3===
| <math>((~))\!</math>
| <math>((~))\!</math>
|}<br>
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
{| 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%"
