Changes

MyWikiBiz, Author Your Legacy — Friday January 10, 2025
Jump to navigationJump to search

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

Revision as of 16:48, 28 May 2008

8,570 bytes added ,  16:48, 28 May 2008
typographical monkey business
Line 2,312: Line 2,312:  
=Work Area 2=
 
=Work Area 2=
   −
===Propositional Forms on Two Variables===
+
==Typographical Monkey Business==
 +
 
 +
===Version 1===
 +
 
 +
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&nbsp;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%"
 +
|+ '''Table 1.  Propositional Forms on Two Variables'''
 +
|- style="background:ghostwhite"
 +
| style="width:16%" | <math>\mathcal{L}_1</math>
 +
| style="width:16%" | <math>\mathcal{L}_2</math>
 +
| style="width:16%" | <math>\mathcal{L}_3</math>
 +
| style="width:16%" | <math>\mathcal{L}_4</math>
 +
| style="width:16%" | <math>\mathcal{L}_5</math>
 +
| style="width:16%" | <math>\mathcal{L}_6</math>
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>x\!</math> :
 +
| 1 1 0 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>y\!</math> :
 +
| 1 0 1 0
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_{0}\!</math>
 +
| <math>f_{0000}\!</math>
 +
| 0 0 0 0
 +
| <math>(\!|~|\!)</math>
 +
| false
 +
| <math>0\!</math>
 +
|-
 +
| <math>f_{1}\!</math>
 +
| <math>f_{0001}\!</math>
 +
| 0 0 0 1
 +
| <math>(\!|x|\!)(\!|y|\!)</math>
 +
| neither x nor y
 +
| <math>\lnot x \land \lnot y</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_{3}\!</math>
 +
| <math>f_{0011}\!</math>
 +
| 0 0 1 1
 +
| <math>(\!|x|\!)</math>
 +
| not x
 +
| <math>\lnot 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_{5}\!</math>
 +
| <math>f_{0101}\!</math>
 +
| 0 1 0 1
 +
| <math>(\!|y|\!)</math>
 +
| not y
 +
| <math>\lnot y</math>
 +
|-
 +
| <math>f_{6}\!</math>
 +
| <math>f_{0110}\!</math>
 +
| 0 1 1 0
 +
| <math>(\!|x,\ y|\!)</math>
 +
| x not equal to y
 +
| <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_{8}\!</math>
 +
| <math>f_{1000}\!</math>
 +
| 1 0 0 0
 +
| <math>x\ y</math>
 +
| x and y
 +
| <math>x \land y</math>
 +
|-
 +
| <math>f_{9}\!</math>
 +
| <math>f_{1001}\!</math>
 +
| 1 0 0 1
 +
| <math>(\!|(\!|x,\ y|\!)|\!)</math>
 +
| x equal to y
 +
| <math>x = y\!</math>
 +
|-
 +
| <math>f_{10}\!</math>
 +
| <math>f_{1010}\!</math>
 +
| 1 0 1 0
 +
| <math>y\!</math>
 +
| y
 +
| <math>y\!</math>
 +
|-
 +
| <math>f_{11}\!</math>
 +
| <math>f_{1011}\!</math>
 +
| 1 0 1 1
 +
| <math>(\!|x\ (\!|y|\!)|\!)</math>
 +
| not x without y
 +
| <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_{1101}\!</math>
 +
| 1 1 0 1
 +
| <math>(\!|(\!|x|\!)\ y|\!)</math>
 +
| not y without x
 +
| <math>x \Leftarrow y</math>
 +
|-
 +
| <math>f_{14}\!</math>
 +
| <math>f_{1110}\!</math>
 +
| 1 1 1 0
 +
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
 +
| x or y
 +
| <math>x \lor y</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| 1 1 1 1
 +
| <math>(\!|(\!|~|\!)|\!)</math>
 +
| true
 +
| <math>1\!</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%"
 +
|+ '''Table 2.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 +
|- style="background:ghostwhite"
 +
| style="width:16%" | &nbsp;
 +
| style="width:16%" | <math>f\!</math>
 +
| style="width:16%" | <math>\operatorname{E}f|_{xy}</math>
 +
| style="width:16%" | <math>\operatorname{E}f|_{x(\!|y|\!)}</math>
 +
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)y}</math>
 +
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
 +
|-
 +
| <math>f_{0}\!</math>
 +
| <math>(\!|~|\!)</math>
 +
| <math>(\!|~|\!)</math>
 +
| <math>(\!|~|\!)</math>
 +
| <math>(\!|~|\!)</math>
 +
| <math>(\!|~|\!)</math>
 +
|-
 +
| <math>f_{1}\!</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_{2}\!</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_{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>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
 +
|-
 +
| <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>
 +
|-
 +
| <math>f_{3}\!</math>
 +
| <math>(\!|x|\!)</math>
 +
| <math>\operatorname{d}x</math>
 +
| <math>\operatorname{d}x</math>
 +
| <math>(\!|\operatorname{d}x|\!)</math>
 +
| <math>(\!|\operatorname{d}x|\!)</math>
 +
|-
 +
| <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>
 +
|-
 +
| <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>
 +
|-
 +
| <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>
 +
|-
 +
| <math>f_{5}\!</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_{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>
 +
|-
 +
| <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>(\!|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_{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>
 +
|-
 +
| <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>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>(\!|(\!|~|\!)|\!)</math>
 +
| <math>(\!|(\!|~|\!)|\!)</math>
 +
| <math>(\!|(\!|~|\!)|\!)</math>
 +
| <math>(\!|(\!|~|\!)|\!)</math>
 +
| <math>(\!|(\!|~|\!)|\!)</math>
 +
|}<br>
 +
 
 +
===Version 2===
    
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.
 
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.
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/67463"

Navigation menu