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__TOC__
==Boolean Functions on Two Variables==
===Boolean Functions, Entitative Graphs, Existential Graphs===
====PNG====
{| align="center" style="font-size:larger; text-align:center; width:100%"
| height="20px" | <math>\text{Boolean Functions on Two Variables}</math>
|-
| [[File:Boolean Functions on Two Variables.png|700px]]
|}
====HTML + JPG====
<br>
<table align="center" cellpadding="10" style="border:2px solid black; font-size:medium; text-align:center;">
<caption style="height:2em;">
<math>\text{Boolean Functions on Two Variables}</math></caption>
<tr style="height:2em;">
<th style="border-bottom:2px solid black;"><math>\text{Boolean Function}</math></th>
<th style="border-bottom:2px solid black;"><math>\text{Entitative Graph}</math></th>
<th style="border-bottom:2px solid black;"><math>\text{Existential Graph}</math></th>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{0}</math>
<br><br><math>\text{false}</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch Root.jpg|32px]]<br><br><math>\text{false}</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch Stem.jpg|32px]]<br><br><math>\text{false}</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{1}</math>
<br><br><math>\text{neither}~ x ~\text{nor}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (XY).jpg|32px]]<br><br><math>\lnot (x \lor y)</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X)(Y).jpg|64px]]<br><br><math>\lnot x \land \lnot y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{2}</math>
<br><br><math>y ~\text{and not}~ x</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X(Y)).jpg|64px]]<br><br><math>\lnot x \land y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X)Y.jpg|40px]]<br><br><math>\lnot x \land y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{3}</math>
<br><br><math>\text{not}~ x</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X).jpg|32px]]<br><br><math>\lnot x</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X).jpg|32px]]<br><br><math>\lnot x</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{4}</math>
<br><br><math>x ~\text{and not}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch ((X)Y).jpg|64px]]<br><br><math>x \land \lnot y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch X(Y).jpg|40px]]<br><br><math>x \land \lnot y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{5}</math>
<br><br><math>\text{not}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (Y).jpg|32px]]<br><br><math>\lnot y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (Y).jpg|32px]]<br><br><math>\lnot y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{6}</math>
<br><br><math>x ~\text{not equal to}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch ((X,Y)).jpg|64px]]<br><br><math>x \ne y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X,Y).jpg|64px]]<br><br><math>x \ne y</math></td>
</tr>
<tr style="height:100px;">
<td style="vertical-align:bottom;border-bottom:2px solid black;"><math>f_{7}</math>
<br><br><math>\text{not both}~ x ~\text{and}~ y</math></td>
<td style="vertical-align:bottom;border-bottom:2px solid black;">
[[File:Cactus Patch (X)(Y).jpg|64px]]<br><br><math>\lnot x \lor \lnot y</math></td>
<td style="vertical-align:bottom;border-bottom:2px solid black;">
[[File:Cactus Patch (XY).jpg|32px]]<br><br><math>\lnot (x \land y)</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{8}</math>
<br><br><math>x ~\text{and}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch ((X)(Y)).jpg|64px]]<br><br><math>x \land y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch XY.jpg|32px]]<br><br><math>x \land y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{9}</math>
<br><br><math>x ~\text{equal to}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X,Y).jpg|64px]]<br><br><math>x = y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch ((X,Y)).jpg|64px]]<br><br><math>x = y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{10}</math>
<br><br><math>y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch Y.jpg|32px]]<br><br><math>y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch Y.jpg|32px]]<br><br><math>y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{11}</math>
<br><br><math>\text{if}~ x ~\text{then}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X)Y.jpg|40px]]<br><br><math>x \Rightarrow y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch (X(Y)).jpg|64px]]<br><br><math>x \Rightarrow y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{12}</math>
<br><br><math>x</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch X.jpg|32px]]<br><br><math>x</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch X.jpg|32px]]<br><br><math>x</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{13}</math>
<br><br><math>\text{if}~ y ~\text{then}~ x</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch X(Y).jpg|40px]]<br><br><math>x \Leftarrow y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch ((X)Y).jpg|64px]]<br><br><math>x \Leftarrow y</math></td>
</tr>
<tr style="height:100px;">
<td style="border-bottom:1px solid black;vertical-align:bottom;"><math>f_{14}</math>
<br><br><math>x ~\text{or}~ y</math></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch XY.jpg|32px]]<br><br><math>x \lor y</math><br></td>
<td style="border-bottom:1px solid black;vertical-align:bottom;">
[[File:Cactus Patch ((X)(Y)).jpg|64px]]<br><br><math>x \lor y</math></td>
</tr>
<tr style="height:100px;">
<td style="vertical-align:bottom;"><math>f_{15}</math>
<br><br><math>\text{true}</math></td>
<td style="vertical-align:bottom;">
[[File:Cactus Patch Stem.jpg|32px]]<br><br><math>\text{true}</math></td>
<td style="vertical-align:bottom;">
[[File:Cactus Patch Root.jpg|32px]]<br><br><math>\text{true}</math></td>
</tr>
</table>
<br>
===Boolean Functions, Truth Tables, Existential Graphs, Logical Formulas===
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px; font-size:medium" | <math>\text{Propositional Forms on Two Variables}</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
|
|
|
|- style="background:ghostwhite"
|
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
|
|
|
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>(~)</math>
| <math>\text{false}</math>
| <math>0</math>
|-
| <math>f_{1}</math>
| <math>f_{0001}</math>
| <math>0~0~0~1</math>
| <math>(x)(y)</math>
| <math>\text{neither}~ x ~\text{nor}~ y</math>
| <math>\lnot x \land \lnot y</math>
|-
| <math>f_{2}</math>
| <math>f_{0010}</math>
| <math>0~0~1~0</math>
| <math>(x)\ y</math>
| <math>y ~\text{without}~ x</math>
| <math>\lnot x \land y</math>
|-
| <math>f_{3}</math>
| <math>f_{0011}</math>
| <math>0~0~1~1</math>
| <math>(x)</math>
| <math>\text{not}~ x</math>
| <math>\lnot x</math>
|-
| <math>f_{4}</math>
| <math>f_{0100}</math>
| <math>0~1~0~0</math>
| <math>x\ (y)</math>
| <math>x ~\text{without}~ y</math>
| <math>x \land \lnot y</math>
|-
| <math>f_{5}</math>
| <math>f_{0101}</math>
| <math>0~1~0~1</math>
| <math>(y)</math>
| <math>\text{not}~ y</math>
| <math>\lnot y</math>
|-
| <math>f_{6}</math>
| <math>f_{0110}</math>
| <math>0~1~1~0</math>
| <math>(x, y)</math>
| <math>x ~\text{not equal to}~ y</math>
| <math>x \ne y</math>
|-
| <math>f_{7}</math>
| <math>f_{0111}</math>
| <math>0~1~1~1</math>
| <math>(x\ y)</math>
| <math>\text{not both}~ x ~\text{and}~ y</math>
| <math>\lnot x \lor \lnot y</math>
|-
| <math>f_{8}</math>
| <math>f_{1000}</math>
| <math>1~0~0~0</math>
| <math>x\ y</math>
| <math>x ~\text{and}~ y</math>
| <math>x \land y</math>
|-
| <math>f_{9}</math>
| <math>f_{1001}</math>
| <math>1~0~0~1</math>
| <math>((x, y))</math>
| <math>x ~\text{equal to}~ y</math>
| <math>x = y</math>
|-
| <math>f_{10}</math>
| <math>f_{1010}</math>
| <math>1~0~1~0</math>
| <math>y</math>
| <math>y</math>
| <math>y</math>
|-
| <math>f_{11}</math>
| <math>f_{1011}</math>
| <math>1~0~1~1</math>
| <math>(x\ (y))</math>
| <math>\text{not}~ x ~\text{without}~ y</math>
| <math>x \Rightarrow y</math>
|-
| <math>f_{12}</math>
| <math>f_{1100}</math>
| <math>1~1~0~0</math>
| <math>x</math>
| <math>x</math>
| <math>x</math>
|-
| <math>f_{13}</math>
| <math>f_{1101}</math>
| <math>1~1~0~1</math>
| <math>((x)\ y)</math>
| <math>\text{not}~ y ~\text{without}~ x</math>
| <math>x \Leftarrow y</math>
|-
| <math>f_{14}</math>
| <math>f_{1110}</math>
| <math>1~1~1~0</math>
| <math>((x)(y))</math>
| <math>x ~\text{or}~ y</math>
| <math>x \lor y</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>((~))</math>
| <math>\text{true}</math>
| <math>1</math>
|}
<br>
