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9,501 bytes added ,  12:46, 14 December 2008
→‎Tables: back to ordinary parentheses for function notation
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==Functional Quantifiers==
 
==Functional Quantifiers==
   −
The '''umpire measure''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0.  Expressed in symbolic form:
+
The '''umpire measure''' of type <math>\Upsilon : (X \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : X \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0.  Expressed in symbolic form:
    
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| <math>\Upsilon \langle u \rangle = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.</math>
+
| <math>\Upsilon (u) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{X \to \mathbb{B}}.</math>
 
|}
 
|}
   −
The '''umpire operator''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> links pairs of propositions in which the first implies the second to a value of 1 and every other pair to a value of 0.  Expressed in symbolic form:
+
The '''umpire operator''' of type <math>\Upsilon : (X \to \mathbb{B})^2 \to \mathbb{B}</math> links pairs of propositions in which the first implies the second to a value of 1 and every other pair to a value of 0.  Expressed in symbolic form:
    
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| <math>\Upsilon \langle u, v \rangle = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
+
| <math>\Upsilon (u, v) = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
 
|}
 
|}
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 1 \ldots 15,</math>
+
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math>
 
|}
 
|}
   Line 452: Line 452:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>
+
| <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f),</math>
 
|-
 
|-
| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</math>
+
| <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i).</math>
 
|}
 
|}
 +
 +
====Table 1====
    
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;1.  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.
 
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;1.  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.
    
{| 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 1.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
+
|+ '''Table 1.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
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| style="background:black; color:white" | 1
 
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 2====
    
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;2.  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.
 
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;2.  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 2.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''
+
|+ '''Table 2.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
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| <math>f_{14}</math>
 
| <math>f_{14}</math>
 
| 1110
 
| 1110
| <math>((u)(y))\!</math>
+
| <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
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| style="background:black; color:white" | 1
 
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 3====
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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| 0000
 
| 0000
 
| <math>(~)</math>
 
| <math>(~)</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_1</math>
 
| <math>f_1</math>
 
| 0001
 
| 0001
 
| <math>(u)(v)\!</math>
 
| <math>(u)(v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_2</math>
 
| <math>f_2</math>
 
| 0010
 
| 0010
 
| <math>(u) v\!</math>
 
| <math>(u) v\!</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_3</math>
 
| <math>f_3</math>
 
| 0011
 
| 0011
 
| <math>(u)\!</math>
 
| <math>(u)\!</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_4</math>
 
| <math>f_4</math>
 
| 0100
 
| 0100
 
| <math>u (v)\!</math>
 
| <math>u (v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_5</math>
 
| <math>f_5</math>
 
| 0101
 
| 0101
 
| <math>(v)\!</math>
 
| <math>(v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_6</math>
 
| <math>f_6</math>
 
| 0110
 
| 0110
 
| <math>(u, v)\!</math>
 
| <math>(u, v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_7</math>
 
| <math>f_7</math>
 
| 0111
 
| 0111
 
| <math>(u v)\!</math>
 
| <math>(u v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_8</math>
 
| <math>f_8</math>
 
| 1000
 
| 1000
 
| <math>u v\!</math>
 
| <math>u v\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_9</math>
 
| <math>f_9</math>
 
| 1001
 
| 1001
 
| <math>((u, v))\!</math>
 
| <math>((u, v))\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{10}</math>
 
| <math>f_{10}</math>
 
| 1010
 
| 1010
 
| <math>v\!</math>
 
| <math>v\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{11}</math>
 
| <math>f_{11}</math>
 
| 1011
 
| 1011
 
| <math>(u (v))\!</math>
 
| <math>(u (v))\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{12}</math>
 
| <math>f_{12}</math>
 
| 1100
 
| 1100
 
| <math>u\!</math>
 
| <math>u\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{13}</math>
 
| <math>f_{13}</math>
 
| 1101
 
| 1101
 
| <math>((u) v)\!</math>
 
| <math>((u) v)\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{14}</math>
 
| <math>f_{14}</math>
 
| 1110
 
| 1110
 
| <math>((u)(v))\!</math>
 
| <math>((u)(v))\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{15}</math>
 
| <math>f_{15}</math>
 
| 1111
 
| 1111
 
| <math>((~))</math>
 
| <math>((~))</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 4====
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Line 1,375: Line 1,383:  
| 0000
 
| 0000
 
| <math>(~)</math>
 
| <math>(~)</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_1</math>
 
| <math>f_1</math>
 
| 0001
 
| 0001
 
| <math>(u)(v)\!</math>
 
| <math>(u)(v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_2</math>
 
| <math>f_2</math>
 
| 0010
 
| 0010
 
| <math>(u) v\!</math>
 
| <math>(u) v\!</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_4</math>
 
| <math>f_4</math>
 
| 0100
 
| 0100
 
| <math>u (v)\!</math>
 
| <math>u (v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_8</math>
 
| <math>f_8</math>
 
| 1000
 
| 1000
 
| <math>u v\!</math>
 
| <math>u v\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_3</math>
 
| <math>f_3</math>
 
| 0011
 
| 0011
 
| <math>(u)\!</math>
 
| <math>(u)\!</math>
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_{12}</math>
 
| <math>f_{12}</math>
 
| 1100
 
| 1100
 
| <math>u\!</math>
 
| <math>u\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_6</math>
 
| <math>f_6</math>
 
| 0110
 
| 0110
 
| <math>(u, v)\!</math>
 
| <math>(u, v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_9</math>
 
| <math>f_9</math>
 
| 1001
 
| 1001
 
| <math>((u, v))\!</math>
 
| <math>((u, v))\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_5</math>
 
| <math>f_5</math>
 
| 0101
 
| 0101
 
| <math>(v)\!</math>
 
| <math>(v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_{10}</math>
 
| <math>f_{10}</math>
 
| 1010
 
| 1010
 
| <math>v\!</math>
 
| <math>v\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_7</math>
 
| <math>f_7</math>
 
| 0111
 
| 0111
 
| <math>(u v)\!</math>
 
| <math>(u v)\!</math>
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
 
|-
 
|-
 
| <math>f_{11}</math>
 
| <math>f_{11}</math>
 
| 1011
 
| 1011
 
| <math>(u (v))\!</math>
 
| <math>(u (v))\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{13}</math>
 
| <math>f_{13}</math>
 
| 1101
 
| 1101
 
| <math>((u) v)\!</math>
 
| <math>((u) v)\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{14}</math>
 
| <math>f_{14}</math>
 
| 1110
 
| 1110
 
| <math>((u)(v))\!</math>
 
| <math>((u)(v))\!</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|-
 
|-
 
| <math>f_{15}</math>
 
| <math>f_{15}</math>
 
| 1111
 
| 1111
 
| <math>((~))</math>
 
| <math>((~))</math>
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| &nbsp;
+
| style="background:white; color:black" | 0
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
| 1
+
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 5====
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Line 1,628: Line 1,638:  
| <math>\text{Some}\ u\ \text{is}\ v</math>
 
| <math>\text{Some}\ u\ \text{is}\ v</math>
 
| &nbsp;
 
| &nbsp;
| <math>\text{Some}\ u\ \text{is}\ y</math>
+
| <math>\text{Some}\ u\ \text{is}\ v</math>
 
| <math>\ell_{11}\!</math>
 
| <math>\ell_{11}\!</math>
 
|}<br>
 
|}<br>
12,080

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