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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)

Revision as of 14:00, 13 December 2008

19,652 bytes added ,  14:00, 13 December 2008
redo tables
Line 428: Line 428:  
An acquaintance with the functions of the umpire operator can be gained from Tables 4 and 5, where the 2-dimensional case is worked out in full.
 
An acquaintance with the functions of the umpire operator can be gained from Tables 4 and 5, where the 2-dimensional case is worked out in full.
   −
The auxiliary notations:
+
Define two families of measures:
   −
: <math>\alpha_i f = \Upsilon (f_i, f),\!</math>
+
{| align="center" cellpadding="8"
 +
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math>
 +
|}
   −
: <math>\beta_i f = \Upsilon (f, f_i),\!</math>
+
by means of the following formulas:
   −
define two series of measures:
+
{| align="center" cellpadding="8"
 +
| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>
 +
|-
 +
| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</math>
 +
|}
   −
: <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math>
+
The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;4.  Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.
−
 
  −
incidentally providing compact names for the column headings of the next two Tables.
      
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 4.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>'''
+
|+ '''Table 4.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| align="right" | <math>x:</math><br><math>y:</math>
+
| align="right" | <math>u:</math><br><math>v:</math>
 
| 1100<br>1010
 
| 1100<br>1010
 
| <math>f\!</math>
 
| <math>f\!</math>
 +
| <math>\alpha_0</math>
 +
| <math>\alpha_1</math>
 +
| <math>\alpha_2</math>
 +
| <math>\alpha_3</math>
 +
| <math>\alpha_4</math>
 +
| <math>\alpha_5</math>
 +
| <math>\alpha_6</math>
 +
| <math>\alpha_7</math>
 +
| <math>\alpha_8</math>
 +
| <math>\alpha_9</math>
 +
| <math>\alpha_{10}</math>
 +
| <math>\alpha_{11}</math>
 +
| <math>\alpha_{12}</math>
 +
| <math>\alpha_{13}</math>
 +
| <math>\alpha_{14}</math>
 
| <math>\alpha_{15}</math>
 
| <math>\alpha_{15}</math>
−
| <math>\alpha_{14}</math>
  −
| <math>\alpha_{13}</math>
  −
| <math>\alpha_{12}</math>
  −
| <math>\alpha_{11}</math>
  −
| <math>\alpha_{10}</math>
  −
| <math>\alpha_9</math>
  −
| <math>\alpha_8</math>
  −
| <math>\alpha_7</math>
  −
| <math>\alpha_6</math>
  −
| <math>\alpha_5</math>
  −
| <math>\alpha_4</math>
  −
| <math>\alpha_3</math>
  −
| <math>\alpha_2</math>
  −
| <math>\alpha_1</math>
  −
| <math>\alpha_0</math>
   
|-
 
|-
−
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0000
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_1</math> || 0001 || <math>(x)(y)\!</math>
+
| <math>f_1</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0001
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_2</math> || 0010 || <math>(x) y\!</math>
+
| <math>f_2</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0010
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_3</math> || 0011 || <math>(x)\!</math>
+
| <math>f_3</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0011
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
| <math>(u)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_4</math> || 0100 || <math>x (y)\!</math>
+
| <math>f_4</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_5</math> || 0101 || <math>(y)\!</math>
+
| <math>f_5</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0101
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_6</math> || 0110 || <math>(x, y)\!</math>
+
| <math>f_6</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0110
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_7</math> || 0111 || <math>(x y)\!</math>
+
| <math>f_7</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0111
−
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
| <math>(u v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_8</math> || 1000 || <math>x y\!</math>
+
| <math>f_8</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 1000
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>u v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_9</math> || 1001 || <math>((x, y))\!</math>
+
| <math>f_9</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| 1001
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>((u, v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{10}</math> || 1010 || <math>y\!</math>
+
| <math>f_{10}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| 1010
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{11}</math> || 1011 || <math>(x (y))\!</math>
+
| <math>f_{11}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1    || 1    || 1
+
| 1011
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1    || 1    || 1
+
| <math>(u (v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{12}</math> || 1100 || <math>x\!</math>
+
| <math>f_{12}</math>
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 1100
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>u\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{13}</math> || 1101 || <math>((x) y)\!</math>
+
| <math>f_{13}</math>
−
|&nbsp;||&nbsp;|| 1   || 1    ||&nbsp;||&nbsp;|| 1   || 1
+
| 1101
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>((u) v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{14}</math> || 1110 || <math>((x)(y))\!</math>
+
| <math>f_{14}</math>
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 1110
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>((u)(v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{15}</math> || 1111 || <math>((~))</math>
+
| <math>f_{15}</math>
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 1111
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>((~))</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
The values of the sixteen <math>\beta_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;5.  Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1\!</math> says that <math>f\!</math> is ''below or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\le f_i\!</math> in the implication ordering.
    
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 5.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>'''
+
|+ '''Table 5.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| align="right" | <math>x:</math><br><math>y:</math>
+
| align="right" | <math>u:</math><br><math>v:</math>
 
| 1100<br>1010
 
| 1100<br>1010
 
| <math>f\!</math>
 
| <math>f\!</math>
Line 551: Line 813:  
| <math>\beta_{15}</math>
 
| <math>\beta_{15}</math>
 
|-
 
|-
−
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 0000
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_1</math> || 0001 || <math>(x)(y)\!</math>
+
| <math>f_1</math>
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 0001
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u)(v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_2</math> || 0010 || <math>(x) y\!</math>
+
| <math>f_2</math>
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| 0010
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>(u) v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_3</math> || 0011 || <math>(x)\!</math>
+
| <math>f_3</math>
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0011
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_4</math> || 0100 || <math>x (y)\!</math>
+
| <math>f_4</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| <math>u (v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_5</math> || 0101 || <math>(y)\!</math>
+
| <math>f_5</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| 0101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| <math>(v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_6</math> || 0110 || <math>(x, y)\!</math>
+
| <math>f_6</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| 0110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| <math>(u, v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_7</math> || 0111 || <math>(x y)\!</math>
+
| <math>f_7</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_8</math> || 1000 || <math>x y\!</math>
+
| <math>f_8</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1000
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_9</math> || 1001 || <math>((x, y))\!</math>
+
| <math>f_9</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1001
−
|&nbsp;|| 1   ||&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{10}</math> || 1010 || <math>y\!</math>
+
| <math>f_{10}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1010
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{11}</math> || 1011 || <math>(x (y))\!</math>
+
| <math>f_{11}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1011
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{12}</math> || 1100 || <math>x\!</math>
+
| <math>f_{12}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{13}</math> || 1101 || <math>((x) y)\!</math>
+
| <math>f_{13}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{14}</math> || 1110 || <math>((x)(y))\!</math>
+
| <math>f_{14}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| <math>((u)(y))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{15}</math> || 1111 || <math>((~))\!</math>
+
| <math>f_{15}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>((~))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
  
Jon Awbrey
12,080

edits

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