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, 04:12, 29 May 2008→Work Area 3: cleanup workspace
Line 2,579:
Line 2,579:
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"+|+ '''Table 1. Propositional Forms on Two Variables'''
−|- style="background:paleturquoise"
−|
−{| align="right" style="background:paleturquoise; text-align:right"
−| u :
−|-
−| v :
−|}
−|
−{| align="center" style="background:paleturquoise; text-align:center"
−| 1100
−|-
−| 1010
−|}
−|
−{| align="center" style="background:paleturquoise; text-align:center"
−| f
−|-
−|
−|}
−|
−{| align="center" style="background:paleturquoise; text-align:center"
−| θf
−|-
−|
−|}
−|
−{| align="center" style="background:paleturquoise; text-align:center"
−| θf
−|-
−|
−|}
−|-
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| f<sub>0</sub>
−|-
−| f<sub>1</sub>
−|-
−| f<sub>2</sub>
−|-
−| f<sub>3</sub>
−|-
−| f<sub>4</sub>
−|-
−| f<sub>5</sub>
−|-
−| f<sub>6</sub>
−|-
−| f<sub>7</sub>
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| 0000
−|-
−| 0001
−|-
−| 0010
−|-
−| 0011
−|-
−| 0100
−|-
−| 0101
−|-
−| 0110
−|-
−| 0111
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| ()
−|-
−| (u)(v)
−|-
−| (u) v
−|-
−| (u)
−|-
−| u (v)
−|-
−| (v)
−|-
−| (u, v)
−|-
−| (u v)
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| (( f , () ))
−|-
−| (( f , (u)(v) ))
−|-
−| (( f , (u) v ))
−|-
−| (( f , (u) ))
−|-
−| (( f , u (v) ))
−|-
−| (( f , (v) ))
−|-
−| (( f , (u, v) ))
−|-
−| (( f , (u v) ))
−|}
−|
−{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
−| f + 1
−|-
−| f + u + v + uv
−|-
−| f + v + uv + 1
−|-
−| f + u
−|-
−| f + u + uv + 1
−|-
−| f + v
−|-
−| f + u + v + 1
−|-
−| f + uv
−|}
−|-
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| f<sub>8</sub>
−|-
−| f<sub>9</sub>
−|-
−| f<sub>10</sub>
−|-
−| f<sub>11</sub>
−|-
−| f<sub>12</sub>
−|-
−| f<sub>13</sub>
−|-
−| f<sub>14</sub>
−|-
−| f<sub>15</sub>
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| 1000
−|-
−| 1001
−|-
−| 1010
−|-
−| 1011
−|-
−| 1100
−|-
−| 1101
−|-
−| 1110
−|-
−| 1111
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| u v
−|-
−| ((u, v))
−|-
−| v
−|-
−| (u (v))
−|-
−| u
−|-
−| ((u) v)
−|-
−| ((u)(v))
−|-
−| (())
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| (( f , u v ))
−|-
−| (( f , ((u, v)) ))
−|-
−| (( f , v ))
−|-
−| (( f , (u (v)) ))
−|-
−| (( f , u ))
−|-
−| (( f , ((u) v) ))
−|-
−| (( f , ((u)(v)) ))
−|-
−| (( f , (()) ))
−|}
−|
−{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
−| f + uv + 1
−|-
−| f + u + v
−|-
−| f + v + 1
−|-
−| f + u + uv
−|-
−| f + u + 1
−|-
−| f + v + uv
−|-
−| f + u + v + uv + 1
−|-
−| f
−|}
−|}
−<br>
==Table 14. Differential Propositions==
−<pre>
−Table 14. Differential Propositions
−o-------o--------o---------o-----------o-------------------o----------o
−| | A : 1 1 0 0 | | | |
−| | dA : 1 0 1 0 | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| f_0 | g_0 | 0 0 0 0 | () | False | 0 |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA |
−| | | | | | |
−| | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA |
−| | | | | | |
−| | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA |
−| | | | | | |
−| | g_8 | 1 0 0 0 | A dA | A and dA | A & dA |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A |
−| | | | | | |
−| f_2 | g_12 | 1 1 0 0 | A | A | A |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA |
−| | | | | | |
−| | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA |
−| | | | | | |
−| | g_10 | 1 0 1 0 | dA | dA | dA |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA |
−| | | | | | |
−| | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA |
−| | | | | | |
−| | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA |
−| | | | | | |
−| | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−| | | | | | |
−| f_3 | g_15 | 1 1 1 1 | (()) | True | 1 |
−| | | | | | |
−o-------o--------o---------o-----------o-------------------o----------o
−</pre>
−{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|+ '''Table 14. Differential Propositions'''
−|- style="background:paleturquoise"
−|
−| align="right" | A :
−| 1 1 0 0
−|
−|
−|
−|- style="background:paleturquoise"
−|
−| align="right" | dA :
−| 1 0 1 0
−|
−|
−|
−|-
−| f<sub>0</sub>
−| g<sub>0</sub>
−| 0 0 0 0
−| ( )
−| False
−| 0
−|-
−|
−| g<sub>1</sub>
−| 0 0 0 1
−| (A)(dA)
−| Neither A nor dA
−| ¬A ∧ ¬dA
−|-
−|
−| g<sub>2</sub>
−| 0 0 1 0
−| (A) dA
−| Not A but dA
−| ¬A ∧ dA
−|-
−|
−| g<sub>4</sub>
−| 0 1 0 0
−| A (dA)
−| A but not dA
−| A ∧ ¬dA
−|-
−|
−| g<sub>8</sub>
−| 1 0 0 0
−| A dA
−| A and dA
−| A ∧ dA
−|-
−| f<sub>1</sub>
−| g<sub>3</sub>
−| 0 0 1 1
−| (A)
−| Not A
−| ¬A
−|-
−| f<sub>2</sub>
−| g<sub>12</sub>
−| 1 1 0 0
−| A
−| A
−| A
−|-
−|
−| g<sub>6</sub>
−| 0 1 1 0
−| (A, dA)
−| A not equal to dA
−| A ≠ dA
−|-
−|
−| g<sub>9</sub>
−| 1 0 0 1
−| ((A, dA))
−| A equal to dA
−| A = dA
−|-
−|
−| g<sub>5</sub>
−| 0 1 0 1
−| (dA)
−| Not dA
−| ¬dA
−|-
−|
−| g<sub>10</sub>
−| 1 0 1 0
−| dA
−| dA
−| dA
−|-
−|
−| g<sub>7</sub>
−| 0 1 1 1
−| (A dA)
−| Not both A and dA
−| ¬A ∨ ¬dA
−|-
−|
−| g<sub>11</sub>
−| 1 0 1 1
−| (A (dA))
−| Not A without dA
−| A → dA
−|-
−|
−| g<sub>13</sub>
−| 1 1 0 1
−| ((A) dA)
−| Not dA without A
−| A ← dA
−|-
−|
−| g<sub>14</sub>
−| 1 1 1 0
−| ((A)(dA))
−| A or dA
−| A ∨ dA
−|-
−| f<sub>3</sub>
−| g<sub>15</sub>
−| 1 1 1 1
−| (( ))
−| True
−| 1
−|}
−<br>
−{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|+ '''Table 14. Differential Propositions'''
−|- style="background:paleturquoise"
−|
−| align="right" | <math>x\!</math> :
−| 1 1 0 0
−|
−|
−|
−|- style="background:paleturquoise"
−|
−| align="right" | dA :
−| 1 0 1 0
−|
−|
−|
−|-
−| f<sub>0</sub>
−| g<sub>0</sub>
−| 0 0 0 0
−| ( )
−| False
−| 0
−|-
−|
−{| style="background:lightcyan"
−|
− <br>
− <br>
− <br>
−
−|}
−|
−{| style="background:lightcyan"
−|
−g<sub>1</sub><br>
−g<sub>2</sub><br>
−g<sub>4</sub><br>
−g<sub>8</sub>
−|}
−|
−{| style="background:lightcyan"
−|
−0 0 0 1<br>
−0 0 1 0<br>
−0 1 0 0<br>
−1 0 0 0
−|}
−|
−{| style="background:lightcyan"
−|
−(A)(dA)<br>
−(A) dA <br>
−A (dA)<br>
−A dA
−|}
−|
−{| style="background:lightcyan"
−|
−Neither A nor dA<br>
−Not A but dA<br>
−A but not dA<br>
−A and dA
−|}
−|
−{| style="background:lightcyan"
−|
−¬A ∧ ¬dA<br>
−¬A ∧ dA<br>
−A ∧ ¬dA<br>
−A ∧ dA
−|}
−|-
−|
−{| style="background:lightcyan"
−|
−f<sub>1</sub><br>
−f<sub>2</sub>
−|}
−|
−{| style="background:lightcyan"
−|
−g<sub>3</sub><br>
−g<sub>12</sub>
−|}
−|
−{| style="background:lightcyan"
−|
−0 0 1 1<br>
−1 1 0 0
−|}
−|
−{| style="background:lightcyan"
−|
−(A)<br>
−A
−|}
−|
−{| style="background:lightcyan"
−|
−Not A<br>
−A
−|}
−|
−{| style="background:lightcyan"
−|
−¬A<br>
−A
−|}
−|-
−|
−{| style="background:lightcyan"
−|
− <br>
−
−|}
−|
−{| style="background:lightcyan"
−|
−g<sub>6</sub><br>
−g<sub>9</sub>
−|}
−|
−{| style="background:lightcyan"
−|
−0 1 1 0<br>
−1 0 0 1
−|}
−|
−{| style="background:lightcyan"
−|
−(A, dA)<br>
−((A, dA))
−|}
−|
−{| style="background:lightcyan"
−|
−A not equal to dA<br>
−A equal to dA
−|}
−|
−{| style="background:lightcyan"
−|
−A ≠ dA<br>
−A = dA
−|}
−|-
−|
−{| style="background:lightcyan"
−|
− <br>
−
−|}
−|
−{| style="background:lightcyan"
−|
−g<sub>5</sub><br>
−g<sub>10</sub>
−|}
−|
−{| style="background:lightcyan"
−|
−0 1 0 1<br>
−1 0 1 0
−|}
−|
−{| style="background:lightcyan"
−|
−(dA)<br>
−dA
−|}
−|
−{| style="background:lightcyan"
−|
−Not dA<br>
−dA
−|}
−|
−{| style="background:lightcyan"
−|
−¬dA<br>
−dA
−|}
−|-
−|
−{| style="background:lightcyan"
−|
− <br>
− <br>
− <br>
−
−|}
−|
−{| style="background:lightcyan"
−|
−g<sub>7</sub><br>
−g<sub>11</sub><br>
−g<sub>13</sub><br>
−g<sub>14</sub>
−|}
−|
−{| style="background:lightcyan"
−|
−0 1 1 1<br>
−1 0 1 1<br>
−1 1 0 1<br>
−1 1 1 0
−|}
−|
−{| style="background:lightcyan"
−|
−(A dA)<br>
−(A (dA))<br>
−((A) dA)<br>
−((A)(dA))
−|}
−|
−{| style="background:lightcyan"
−|
−Not both A and dA<br>
−Not A without dA<br>
−Not dA without A<br>
−A or dA
−|}
−|
−{| style="background:lightcyan"
−|
−¬A ∨ ¬dA<br>
−A → dA<br>
−A ← dA<br>
−A ∨ dA
−|}
−|-
−| f<sub>3</sub>
−| g<sub>15</sub>
−| 1 1 1 1
−| (( ))
−| True
−| 1
−|}<br>
−==Table 27. Thematization of Bivariate Propositions==
−<pre>
−Table 27. Thematization of Bivariate Propositions
−o---------o---------o----------o--------------------o--------------------o
−| u : 1 1 0 0 | f | theta (f) | theta (f) |
−| v : 1 0 1 0 | | | |
−o---------o---------o----------o--------------------o--------------------o
−| | | | | |
−| f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 |
−| | | | | |
−| f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv |
−| | | | | |
−| f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 |
−| | | | | |
−| f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u |
−| | | | | |
−| f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 |
−| | | | | |
−| f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v |
−| | | | | |
−| f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 |
−| | | | | |
−| f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv |
−| | | | | |
−o---------o---------o----------o--------------------o--------------------o
−| | | | | |
−| f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 |
−| | | | | |
−| f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v |
−| | | | | |
−| f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 |
−| | | | | |
−| f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv |
−| | | | | |
−| f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 |
−| | | | | |
−| f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv |
−| | | | | |
−| f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 |
−| | | | | |
−| f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f |
−| | | | | |
−o---------o---------o----------o--------------------o--------------------o
−</pre>
−<br><font face="courier new">
−{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−|+ Table 27. Thematization of Bivariate Propositions
−|- style="background:paleturquoise"
−|
−{| align="right" style="background:paleturquoise; text-align:right"
−| u :
−|-
−| v :
−|}
−|
−{| style="background:paleturquoise"
−| 1100
−|-
−| 1010
−|}
−|
−{| style="background:paleturquoise"
−| f
−|-
−|
−|}
−|
−{| style="background:paleturquoise"
−| θf
−|-
−|
−|}
−|
−{| style="background:paleturquoise"
−| θf
−|-
−|
−|}
−|-
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| f<sub>0</sub>
−|-
−| f<sub>1</sub>
−|-
−| f<sub>2</sub>
−|-
−| f<sub>3</sub>
−|-
−| f<sub>4</sub>
−|-
−| f<sub>5</sub>
−|-
−| f<sub>6</sub>
−|-
−| f<sub>7</sub>
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| 0000
−|-
−| 0001
−|-
−| 0010
−|-
−| 0011
−|-
−| 0100
−|-
−| 0101
−|-
−| 0110
−|-
−| 0111
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| ()
−|-
−| (u)(v)
−|-
−| (u) v
−|-
−| (u)
−|-
−| u (v)
−|-
−| (v)
−|-
−| (u, v)
−|-
−| (u v)
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| (( f , () ))
−|-
−| (( f , (u)(v) ))
−|-
−| (( f , (u) v ))
−|-
−| (( f , (u) ))
−|-
−| (( f , u (v) ))
−|-
−| (( f , (v) ))
−|-
−| (( f , (u, v) ))
−|-
−| (( f , (u v) ))
−|}
−|
−{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
−| f + 1
−|-
−| f + u + v + uv
−|-
−| f + v + uv + 1
−|-
−| f + u
−|-
−| f + u + uv + 1
−|-
−| f + v
−|-
−| f + u + v + 1
−|-
−| f + uv
−|}
−|-
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| f<sub>8</sub>
−|-
−| f<sub>9</sub>
−|-
−| f<sub>10</sub>
−|-
−| f<sub>11</sub>
−|-
−| f<sub>12</sub>
−|-
−| f<sub>13</sub>
−|-
−| f<sub>14</sub>
−|-
−| f<sub>15</sub>
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| 1000
−|-
−| 1001
−|-
−| 1010
−|-
−| 1011
−|-
−| 1100
−|-
−| 1101
−|-
−| 1110
−|-
−| 1111
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| u v
−|-
−| ((u, v))
−|-
−| v
−|-
−| (u (v))
−|-
−| u
−|-
−| ((u) v)
−|-
−| ((u)(v))
−|-
−| (())
−|}
−|
−{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
−| (( f , u v ))
−|-
−| (( f , ((u, v)) ))
−|-
−| (( f , v ))
−|-
−| (( f , (u (v)) ))
−|-
−| (( f , u ))
−|-
−| (( f , ((u) v) ))
−|-
−| (( f , ((u)(v)) ))
−|-
−| (( f , (()) ))
−|}
−|
−{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
−| f + uv + 1
−|-
−| f + u + v
−|-
−| f + v + 1
−|-
−| f + u + uv
−|-
−| f + u + 1
−|-
−| f + v + uv
−|-
−| f + u + v + uv + 1
−|-
−| f
−|}
−|}
−</font><br>
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.
+
+===Variant 1===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Line 2,717:
Line 2,719:
|}<br>
|}<br>
−===Variant 2===
−{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
Line 3,175:
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| 1
| 1
|}<br>
|}<br>
+
+===Variant 3===
+
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ '''Table 1. Propositional Forms on Two Variables'''
+|- style="background:paleturquoise"
+|
+{| align="right" style="background:paleturquoise; text-align:right"
+| u :
+|-
+| v :
+|}
+|
+{| align="center" style="background:paleturquoise; text-align:center"
+| 1100
+|-
+| 1010
+|}
+|
+{| align="center" style="background:paleturquoise; text-align:center"
+| f
+|-
+|
+|}
+|
+{| align="center" style="background:paleturquoise; text-align:center"
+| θf
+|-
+|
+|}
+|
+{| align="center" style="background:paleturquoise; text-align:center"
+| θf
+|-
+|
+|}
+|-
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| f<sub>0</sub>
+|-
+| f<sub>1</sub>
+|-
+| f<sub>2</sub>
+|-
+| f<sub>3</sub>
+|-
+| f<sub>4</sub>
+|-
+| f<sub>5</sub>
+|-
+| f<sub>6</sub>
+|-
+| f<sub>7</sub>
+|}
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| 0000
+|-
+| 0001
+|-
+| 0010
+|-
+| 0011
+|-
+| 0100
+|-
+| 0101
+|-
+| 0110
+|-
+| 0111
+|}
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| ()
+|-
+| (u)(v)
+|-
+| (u) v
+|-
+| (u)
+|-
+| u (v)
+|-
+| (v)
+|-
+| (u, v)
+|-
+| (u v)
+|}
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| (( f , () ))
+|-
+| (( f , (u)(v) ))
+|-
+| (( f , (u) v ))
+|-
+| (( f , (u) ))
+|-
+| (( f , u (v) ))
+|-
+| (( f , (v) ))
+|-
+| (( f , (u, v) ))
+|-
+| (( f , (u v) ))
+|}
+|
+{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
+| f + 1
+|-
+| f + u + v + uv
+|-
+| f + v + uv + 1
+|-
+| f + u
+|-
+| f + u + uv + 1
+|-
+| f + v
+|-
+| f + u + v + 1
+|-
+| f + uv
+|}
+|-
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| f<sub>8</sub>
+|-
+| f<sub>9</sub>
+|-
+| f<sub>10</sub>
+|-
+| f<sub>11</sub>
+|-
+| f<sub>12</sub>
+|-
+| f<sub>13</sub>
+|-
+| f<sub>14</sub>
+|-
+| f<sub>15</sub>
+|}
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| 1000
+|-
+| 1001
+|-
+| 1010
+|-
+| 1011
+|-
+| 1100
+|-
+| 1101
+|-
+| 1110
+|-
+| 1111
+|}
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| u v
+|-
+| ((u, v))
+|-
+| v
+|-
+| (u (v))
+|-
+| u
+|-
+| ((u) v)
+|-
+| ((u)(v))
+|-
+| (())
+|}
+|
+{| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
+| (( f , u v ))
+|-
+| (( f , ((u, v)) ))
+|-
+| (( f , v ))
+|-
+| (( f , (u (v)) ))
+|-
+| (( f , u ))
+|-
+| (( f , ((u) v) ))
+|-
+| (( f , ((u)(v)) ))
+|-
+| (( f , (()) ))
+|}
+|
+{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
+| f + uv + 1
+|-
+| f + u + v
+|-
+| f + v + 1
+|-
+| f + u + uv
+|-
+| f + u + 1
+|-
+| f + v + uv
+|-
+| f + u + v + uv + 1
+|-
+| f
+|}
+|}
+<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.
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.
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o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
</pre>
</pre>
−
