MyWikiBiz

Changes

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

Revision as of 16:58, 30 May 2008

2,655 bytes added ,  16:58, 30 May 2008
→‎Propositional Forms on Two Variables: tabula razzula dazzula
Line 3,063: Line 3,063:  
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
 
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
   −
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 1.  Propositional Forms on Two Variables'''
+
|+ '''Table 2.  Propositional Forms on Two Variables'''
−
|- style="background:ghostwhite"
+
|- style="background:ghostwhite; height:36px"
 
| <math>\mathcal{L}_1</math>
 
| <math>\mathcal{L}_1</math>
 
| <math>\mathcal{L}_2</math>
 
| <math>\mathcal{L}_2</math>
Line 3,072: Line 3,072:  
| <math>\mathcal{L}_5</math>
 
| <math>\mathcal{L}_5</math>
 
| <math>\mathcal{L}_6</math>
 
| <math>\mathcal{L}_6</math>
−
|- style="background:ghostwhite"
+
|- style="background:ghostwhite; height:48px"
 
| &nbsp;
 
| &nbsp;
−
| align="right" | <math>x\!</math> :
+
|
−
| 1 1 0 0
+
{| align="right" style="background:ghostwhite; text-align:right"
−
| &nbsp;
+
|-
−
| &nbsp;
+
| <math>x\!</math> :
−
| &nbsp;
+
|-
−
|- style="background:ghostwhite"
+
| <math>y\!</math> :
−
| &nbsp;
+
|}
−
| align="right" | <math>y\!</math> :
+
|
 +
{| align="center" style="background:ghostwhite"
 +
|-
 +
| 1 1 0 0
 +
|-
 
| 1 0 1 0
 
| 1 0 1 0
 +
|}
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
|-
 
|-
−
| <math>f_{0}\!</math>
+
|
−
| <math>f_{0000}\!</math>
+
{| align="center"
−
| 0 0 0 0
+
|-
−
| <math>(~)\!</math>
+
| height="36px" | <p><math>f_{0}\!</math></p>
−
| <math>\operatorname{false}</math>
+
|-
−
| <math>0\!</math>
+
| height="36px" | <p><math>f_{1}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{2}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{3}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{4}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{5}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{6}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{7}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>f_{0000}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{0001}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{0010}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{0011}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{0100}\!</math></p>
 
|-
 
|-
−
| <math>f_{1}\!</math>
+
| height="36px" | <p><math>f_{0101}\!</math></p>
−
| <math>f_{0001}\!</math>
  −
| 0 0 0 1
  −
| <math>(x)(y)\!</math>
  −
| <math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math>
  −
| <math>\lnot x \land \lnot y\!</math>
   
|-
 
|-
−
| <math>f_{2}\!</math>
+
| height="36px" | <p><math>f_{0110}\!</math></p>
−
| <math>f_{0010}\!</math>
  −
| 0 0 1 0
  −
| <math>(x)\ y\!</math>
  −
| <math>y\ \operatorname{without}\ x</math>
  −
| <math>\lnot x \land y\!</math>
   
|-
 
|-
−
| <math>f_{3}\!</math>
+
| height="36px" | <p><math>f_{0111}\!</math></p>
−
| <math>f_{0011}\!</math>
+
|}
−
| 0 0 1 1
+
|
−
| <math>(x)\!</math>
+
{| align="center"
−
| <math>\operatorname{not}\ x</math>
  −
| <math>\lnot x\!</math>
   
|-
 
|-
−
| <math>f_{4}\!</math>
+
| height="36px" | 0 0 0 0
−
| <math>f_{0100}\!</math>
  −
| 0 1 0 0
  −
| <math>x\ (y)\!</math>
  −
| <math>x\ \operatorname{without}\ y</math>
  −
| <math>x \land \lnot y\!</math>
   
|-
 
|-
−
| <math>f_{5}\!</math>
+
| height="36px" | 0 0 0 1
−
| <math>f_{0101}\!</math>
  −
| 0 1 0 1
  −
| <math>(y)\!</math>
  −
| <math>\operatorname{not}\ y</math>
  −
| <math>\lnot y\!</math>
   
|-
 
|-
−
| <math>f_{6}\!</math>
+
| height="36px" | 0 0 1 0
−
| <math>f_{0110}\!</math>
  −
| 0 1 1 0
  −
| <math>(x,\ y)\!</math>
  −
| <math>x\ \operatorname{not~equal~to}\ y</math>
  −
| <math>x \ne y\!</math>
   
|-
 
|-
−
| <math>f_{7}\!</math>
+
| height="36px" | 0 0 1 1
−
| <math>f_{0111}\!</math>
  −
| 0 1 1 1
  −
| <math>(x\ y)\!</math>
  −
| <math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math>
  −
| <math>\lnot x \lor \lnot y\!</math>
   
|-
 
|-
−
| <math>f_{8}\!</math>
+
| height="36px" | 0 1 0 0
−
| <math>f_{1000}\!</math>
  −
| 1 0 0 0
  −
| <math>x\ y\!</math>
  −
| <math>x\ \operatorname{and}\ y</math>
  −
| <math>x \land y\!</math>
   
|-
 
|-
−
| <math>f_{9}\!</math>
+
| height="36px" | 0 1 0 1
−
| <math>f_{1001}\!</math>
  −
| 1 0 0 1
  −
| <math>((x,\ y))\!</math>
  −
| <math>x\ \operatorname{equal~to}\ y</math>
  −
| <math>x = y\!</math>
   
|-
 
|-
−
| <math>f_{10}\!</math>
+
| height="36px" | 0 1 1 0
−
| <math>f_{1010}\!</math>
+
|-
−
| 1 0 1 0
+
| height="36px" | 0 1 1 1
−
| <math>y\!</math>
+
|}
−
| <math>y\!</math>
+
|
−
| <math>y\!</math>
+
{| align="center"
 +
|-
 +
| height="36px" | <p><math>(~)\!</math></p>
 +
|-
 +
| height="36px" | <p><math>(x)(y)\!</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>(y)\!</math></p>
 +
|-
 +
| height="36px" | <p><math>(x,\ y)\!</math></p>
 +
|-
 +
| height="36px" | <p><math>(x\ y)\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>\operatorname{false}</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>\operatorname{not}\ x</math></p>
 +
|-
 +
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>0\!</math></p>
 +
|-
 +
| height="36px" | <p><math>\lnot x \land \lnot y</math></p>
 +
|-
 +
| height="36px" | <p><math>\lnot x \land y</math></p>
 +
|-
 +
| height="36px" | <p><math>\lnot x</math></p>
 +
|-
 +
| height="36px" | <p><math>x \land \lnot y</math></p>
 +
|-
 +
| height="36px" | <p><math>\lnot y</math></p>
 +
|-
 +
| height="36px" | <p><math>x \ne y</math></p>
 +
|-
 +
| height="36px" | <p><math>\lnot x \lor \lnot y</math></p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>f_{8}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{9}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{10}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{11}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{12}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{13}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{14}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{15}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>f_{1000}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1001}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1010}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1011}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1100}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1101}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1110}\!</math></p>
 +
|-
 +
| height="36px" | <p><math>f_{1111}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | 1 0 0 0
 +
|-
 +
| height="36px" | 1 0 0 1
 +
|-
 +
| height="36px" | 1 0 1 0
 +
|-
 +
| height="36px" | 1 0 1 1
 +
|-
 +
| height="36px" | 1 1 0 0
 +
|-
 +
| height="36px" | 1 1 0 1
 +
|-
 +
| height="36px" | 1 1 1 0
 +
|-
 +
| height="36px" | 1 1 1 1
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| 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\!</math></p>
 +
|-
 +
| height="36px" | <p><math>((x)\ y)\!</math></p>
 +
|-
 +
| height="36px" | <p><math>((x)(y))\!</math></p>
 +
|-
 +
| height="36px" | <p><math>((~))\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>y\!</math></p>
 +
|-
 +
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>x\!</math></p>
 +
|-
 +
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
 +
|-
 +
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
 +
|-
 +
| height="36px" | <p><math>\operatorname{true}</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <p><math>x \land y</math></p>
 +
|-
 +
| height="36px" | <p><math>x = y\!</math></p>
 +
|-
 +
| height="36px" | <p><math>y\!</math></p>
 
|-
 
|-
−
| <math>f_{11}\!</math>
+
| height="36px" | <p><math>x \Rightarrow y</math></p>
−
| <math>f_{1011}\!</math>
  −
| 1 0 1 1
  −
| <math>(x\ (y))\!</math>
  −
| <math>\operatorname{not}\ x\ \operatorname{without}\ y</math>
  −
| <math>x \Rightarrow y\!</math>
   
|-
 
|-
−
| <math>f_{12}\!</math>
+
| height="36px" | <p><math>x\!</math></p>
−
| <math>f_{1100}\!</math>
  −
| 1 1 0 0
  −
| <math>x\!</math>
  −
| <math>x\!</math>
  −
| <math>x\!</math>
   
|-
 
|-
−
| <math>f_{13}\!</math>
+
| height="36px" | <p><math>x \Leftarrow y</math></p>
−
| <math>f_{1101}\!</math>
  −
| 1 1 0 1
  −
| <math>((x)\ y)\!</math>
  −
| <math>\operatorname{not}\ y\ \operatorname{without}\ x</math>
  −
| <math>x \Leftarrow y\!</math>
   
|-
 
|-
−
| <math>f_{14}\!</math>
+
| height="36px" | <p><math>x \lor y</math></p>
−
| <math>f_{1110}\!</math>
  −
| 1 1 1 0
  −
| <math>((x)(y))\!</math>
  −
| <math>x\ \operatorname{or}\ y</math>
  −
| <math>x \lor y\!</math>
   
|-
 
|-
−
| <math>f_{15}\!</math>
+
| height="36px" | <p><math>1\!</math></p>
−
| <math>f_{1111}\!</math>
+
|}
−
| 1 1 1 1
  −
| <math>((~))\!</math>
  −
| <math>\operatorname{true}</math>
  −
| <math>1\!</math>
   
|}
 
|}
 
<br>
 
<br>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/67554"