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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 01:48, 30 May 2008
, 01:48, 30 May 2008→Table 2 : Variant 3: edit table
|+ '''Table 2. Propositional Forms on Two Variables'''
|+ '''Table 2. Propositional Forms on Two Variables'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| <math>\mathcal{L}_1</math>
| <math>\mathcal{L}_2</math>
| <math>\mathcal{L}_3</math>
| <math>\mathcal{L}_4</math>
| <math>\mathcal{L}_5</math>
|-
|
|
{| align="right" style="background:ghostwhite; text-align:right"
{| align="right" style="background:ghostwhite; text-align:right"
| u :
| <math>x\!</math> :
|-
|-
| v :
| <math>y\!</math> :
|}
|}
|
|
{| align="center" style="background:ghostwhite"
{| align="center" style="background:ghostwhite"
| 1100
| 1 1 0 0
|-
|-
| 1010
| 1 0 1 0
|}
|}
|
|
{| align="center" style="background:ghostwhite"
{| align="center" style="background:ghostwhite"
| f
|
|-
|-
|
|
|
|
{| align="center" style="background:ghostwhite"
{| align="center" style="background:ghostwhite"
| θf
|
|-
|-
|
|
|
|
{| align="center" style="background:ghostwhite"
{| align="center" style="background:ghostwhite"
| θf
|
|-
|-
|
|
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| f<sub>0</sub>
| <math>f_{0}\!</math>
|-
|-
| f<sub>1</sub>
| <math>f_{1}\!</math>
|-
|-
| f<sub>2</sub>
| <math>f_{2}\!</math>
|-
|-
| f<sub>3</sub>
| <math>f_{3}\!</math>
|-
|-
| f<sub>4</sub>
| <math>f_{4}\!</math>
|-
|-
| f<sub>5</sub>
| <math>f_{5}\!</math>
|-
|-
| f<sub>6</sub>
| <math>f_{6}\!</math>
|-
|-
| f<sub>7</sub>
| <math>f_{7}\!</math>
|}
|}
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| 0000
| 0 0 0 0
|-
|-
| 0001
| 0 0 0 1
|-
|-
| 0010
| 0 0 1 0
|-
|-
| 0011
| 0 0 1 1
|-
|-
| 0100
| 0 1 0 0
|-
|-
| 0101
| 0 1 0 1
|-
|-
| 0110
| 0 1 1 0
|-
|-
| 0111
| 0 1 1 1
|}
|}
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| ()
| <math>(~)\!</math>
|-
|-
| (u)(v)
| <math>(x)(y)\!</math>
|-
|-
| (u) v
| <math>(x)\ y\!</math>
|-
|-
| (u)
| <math>(x)\!</math>
|-
|-
| u (v)
| <math>x\ (y)\!</math>
|-
|-
| (v)
| <math>(y)\!</math>
|-
|-
| (u, v)
| <math>(x,\ y)\!</math>
|-
|-
| (u v)
| <math>(x\ y)\!</math>
|}
|}
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| (( f , () ))
|
|-
|-
| (( f , (u)(v) ))
|
|-
|-
| (( f , (u) v ))
|
|-
|-
| (( f , (u) ))
|
|-
|-
| (( f , u (v) ))
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|-
|-
| (( f , (v) ))
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|-
|-
| (( f , (u, v) ))
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|-
|-
| (( f , (u v) ))
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|}
|}
|
|
{| align="left" cellpadding="2" style="text-align:left"
{| align="left" cellpadding="2" style="text-align:left"
| f + 1
|
|-
|-
| f + u + v + uv
|
|-
|-
| f + v + uv + 1
|
|-
|-
| f + u
|
|-
|-
| f + u + uv + 1
|
|-
|-
| f + v
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|-
|-
| f + u + v + 1
|
|-
|-
| f + uv
|
|}
|}
|-
|-
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| f<sub>8</sub>
| <math>f_{8}\!</math>
|-
|-
| f<sub>9</sub>
| <math>f_{9}\!</math>
|-
|-
| f<sub>10</sub>
| <math>f_{10}\!</math>
|-
|-
| f<sub>11</sub>
| <math>f_{11}\!</math>
|-
|-
| f<sub>12</sub>
| <math>f_{12}\!</math>
|-
|-
| f<sub>13</sub>
| <math>f_{13}\!</math>
|-
|-
| f<sub>14</sub>
| <math>f_{14}\!</math>
|-
|-
| f<sub>15</sub>
| <math>f_{15}\!</math>
|}
|}
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| 1000
| 1 0 0 0
|-
|-
| 1001
| 1 0 0 1
|-
|-
| 1010
| 1 0 1 0
|-
|-
| 1011
| 1 0 1 1
|-
|-
| 1100
| 1 1 0 0
|-
|-
| 1101
| 1 1 0 1
|-
|-
| 1110
| 1 1 1 0
|-
|-
| 1111
| 1 1 1 1
|}
|}
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| u v
| <math>x\ y\!</math>
|-
|-
| ((u, v))
| <math>((x,\ y))\!</math>
|-
|-
| v
| <math>y\!</math>
|-
|-
| (u (v))
| <math>(x\ (y))\!</math>
|-
|-
| u
| <math>x\!</math>
|-
|-
| ((u) v)
| <math>((x)\ y)\!</math>
|-
|-
| ((u)(v))
| <math>((x)(y))\!</math>
|-
|-
| (())
| <math>((~))\!</math>
|}
|}
|
|
{| align="center" cellpadding="2"
{| align="center" cellpadding="2"
| (( f , u v ))
|
|-
|-
| (( f , ((u, v)) ))
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|-
|-
| (( f , v ))
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|-
|-
| (( f , (u (v)) ))
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|-
|-
| (( f , u ))
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|-
|-
| (( f , ((u) v) ))
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|-
|-
| (( f , ((u)(v)) ))
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|-
|-
| (( f , (()) ))
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|}
|}
|
|
{| align="left" cellpadding="2" style="text-align:left"
{| align="left" cellpadding="2" style="text-align:left"
| f + uv + 1
|
|-
|-
| f + u + v
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|-
|-
| f + v + 1
|
|-
|-
| f + u + uv
|
|-
|-
| f + u + 1
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|-
|-
| f + v + uv
|
|-
|-
| f + u + v + uv + 1
|
|-
|-
| f
|
|}
|}
|}
|}