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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 14:20, 11 July 2008
, 14:20, 11 July 2008→Tables of Propositional Forms: HTML → TeX
<math>\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}</math>
<math>\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| <math>~</math>
| <math>~\!</math>
| align="right" | <math>x\colon\!</math>
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0</math>
| <math>1~1~0~0\!</math>
| <math>~</math>
| <math>~\!</math>
| <math>~</math>
| <math>~\!</math>
| <math>~</math>
| <math>~\!</math>
|-
|-
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| <math>~</math>
| <math>~\!</math>
| align="right" | <math>y\colon\!</math>
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0</math>
| <math>1~0~1~0\!</math>
| <math>~</math>
| <math>~\!</math>
| <math>~</math>
| <math>~\!</math>
| <math>~</math>
| <math>~\!</math>
|-
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0
| <math>f_{0}\!</math>
| <math>f_{0000}\!</math>
| <math>0~0~0~0\!</math>
| <math>(~)\!</math>
| <math>\mbox{false}\!</math>
| <math>0\!</math>
|-
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y
| <math>f_{1}\!</math>
| <math>f_{0001}\!</math>
| <math>0~0~0~1\!</math>
| <math>(x)(y)\!</math>
| <math>\mbox{neither}\ x\ \mbox{nor}\ y\!</math>
| <math>\lnot x \land \lnot y\!</math>
|-
|-
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y
| <math>f_{2}\!</math>
| <math>f_{0010}\!</math>
| <math>0~0~1~0\!</math>
| <math>(x)\ y\!</math>
| <math>y\ \mbox{without}\ x\!</math>
| <math>\lnot x \land y\!</math>
|-
|-
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x
| <math>f_{3}\!</math>
| <math>f_{0011}\!</math>
| <math>0~0~1~1\!</math>
| <math>(x)\!</math>
| <math>\mbox{not}\ x\!</math>
| <math>\lnot x\!</math>
|-
|-
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y
| <math>f_{4}\!</math>
| <math>f_{0100}\!</math>
| <math>0~1~0~0\!</math>
| <math>x\ (y)\!</math>
| <math>x\ \mbox{without}\ y\!</math>
| <math>x \land \lnot y\!</math>
|-
|-
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y
| <math>f_{5}\!</math>
| <math>f_{0101}\!</math>
| <math>0~1~0~1\!</math>
| <math>(y)\!</math>
| <math>\mbox{not}\ y\!</math>
| <math>\lnot y\!</math>
|-
|-
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y
| <math>f_{6}\!</math>
| <math>f_{0110}\!</math>
| <math>0~1~1~0\!</math>
| <math>(x, y)\!</math>
| <math>x\ \mbox{not equal to}\ y\!</math>
| <math>x \ne y\!</math>
|-
|-
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y
| <math>f_{7}\!</math>
| <math>f_{0111}\!</math>
| <math>0~1~1~1\!</math>
| <math>(x\ y)\!</math>
| <math>\mbox{not both}\ x\ \mbox{and}\ y\!</math>
| <math>\lnot x \lor \lnot y\!</math>
|-
|-
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y
| <math>f_{8}\!</math>
| <math>f_{1000}\!</math>
| <math>1~0~0~0\!</math>
| <math>x\ y\!</math>
| <math>x\ \mbox{and}\ y\!</math>
| <math>x \land y\!</math>
|-
|-
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
| <math>f_{9}\!</math>
| <math>f_{1001}\!</math>
| <math>1~0~0~1\!</math>
| <math>((x, y))\!</math>
| <math>x\ \mbox{equal to}\ y\!</math>
| <math>x = y\!</math>
|-
|-
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
| <math>f_{10}\!</math>
| <math>f_{1010}\!</math>
| <math>1~0~1~0\!</math>
| <math>y\!</math>
| <math>y\!</math>
| <math>y\!</math>
|-
|-
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y
| <math>f_{11}\!</math>
| <math>f_{1011}\!</math>
| <math>1~0~1~1\!</math>
| <math>(x\ (y))\!</math>
| <math>\mbox{not}\ x\ \mbox{without}\ y\!</math>
| <math>x \Rightarrow y\!</math>
|-
|-
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
| <math>f_{12}\!</math>
| <math>f_{1100}\!</math>
| <math>1~1~0~0\!</math>
| <math>x\!</math>
| <math>x\!</math>
| <math>x\!</math>
|-
|-
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y
| <math>f_{13}\!</math>
| <math>f_{1101}\!</math>
| <math>1~1~0~1\!</math>
| <math>((x)\ y)\!</math>
| <math>\mbox{not}\ y\ \mbox{without}\ x\!</math>
| <math>x \Leftarrow y\!</math>
|-
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y
| <math>f_{14}\!</math>
| <math>f_{1110}\!</math>
| <math>1~1~1~0\!</math>
| <math>((x)(y))\!</math>
| <math>x\ \mbox{or}\ y\!</math>
| <math>x \lor y\!</math>
|-
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>1~1~1~1\!</math>
| <math>((~))\!</math>
| <math>\mbox{true}\!</math>
| <math>1\!</math>
|}<br>
|}<br>