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| <math>\underline{1}</math> | | <math>\underline{1}</math> | ||
| <math>\underline{((} ~ \underline{))}</math> | | <math>\underline{((} ~ \underline{))}</math> | ||
+ | |} | ||
+ | |||
+ | <br> | ||
+ | |||
+ | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" | ||
+ | |+ '''Table 17. Boolean Functions on Two Variables''' | ||
+ | |- style="background:ghostwhite" | ||
+ | | width="14%" | <math>F\!</math> | ||
+ | | width="14%" | <math>F\!</math> | ||
+ | | colspan="4" | <math>F(x, y)\!</math> | ||
+ | | width="24%" | <math>F\!</math> | ||
+ | |- style="background:ghostwhite" | ||
+ | | width="14%" | | ||
+ | | width="14%" | | ||
+ | | width="12%" | <math>F(\underline{1}, \underline{1})</math> | ||
+ | | width="12%" | <math>F(\underline{1}, \underline{0})</math> | ||
+ | | width="12%" | <math>F(\underline{0}, \underline{1})</math> | ||
+ | | width="12%" | <math>F(\underline{0}, \underline{0})</math> | ||
+ | | width="24%" | | ||
+ | |- | ||
+ | | <math>F_{0}^{(2)}\!</math> | ||
+ | | <math>F_{0000}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>(~)</math> | ||
+ | |- | ||
+ | | <math>F_{1}^{(2)}\!</math> | ||
+ | | <math>F_{0001}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>(x)(y)\!</math> | ||
+ | |- | ||
+ | | <math>F_{2}^{(2)}\!</math> | ||
+ | | <math>F_{0010}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>(x) y\!</math> | ||
+ | |- | ||
+ | | <math>F_{3}^{(2)}\!</math> | ||
+ | | <math>F_{0011}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>(x)\!</math> | ||
+ | |- | ||
+ | | <math>F_{4}^{(2)}\!</math> | ||
+ | | <math>F_{0100}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>x (y)\!</math> | ||
+ | |- | ||
+ | | <math>F_{5}^{(2)}\!</math> | ||
+ | | <math>F_{0101}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>(y)\!</math> | ||
+ | |- | ||
+ | | <math>F_{6}^{(2)}\!</math> | ||
+ | | <math>F_{0110}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>(x, y)\!</math> | ||
+ | |- | ||
+ | | <math>F_{7}^{(2)}\!</math> | ||
+ | | <math>F_{0111}^{(2)}\!</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>(x y)\!</math> | ||
+ | |- | ||
+ | | <math>F_{8}^{(2)}\!</math> | ||
+ | | <math>F_{1000}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>x y\!</math> | ||
+ | |- | ||
+ | | <math>F_{9}^{(2)}\!</math> | ||
+ | | <math>F_{1001}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>((x, y))\!</math> | ||
+ | |- | ||
+ | | <math>F_{10}^{(2)}\!</math> | ||
+ | | <math>F_{1010}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>y\!</math> | ||
+ | |- | ||
+ | | <math>F_{11}^{(2)}\!</math> | ||
+ | | <math>F_{1011}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>(x (y))\!</math> | ||
+ | |- | ||
+ | | <math>F_{12}^{(2)}\!</math> | ||
+ | | <math>F_{1100}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>x\!</math> | ||
+ | |- | ||
+ | | <math>F_{13}^{(2)}\!</math> | ||
+ | | <math>F_{1101}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>((x)y)\!</math> | ||
+ | |- | ||
+ | | <math>F_{14}^{(2)}\!</math> | ||
+ | | <math>F_{1110}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{0}</math> | ||
+ | | <math>((x)(y))\!</math> | ||
+ | |- | ||
+ | | <math>F_{15}^{(2)}\!</math> | ||
+ | | <math>F_{1111}^{(2)}\!</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>\underline{1}</math> | ||
+ | | <math>((~))</math> | ||
|} | |} | ||
Revision as of 18:30, 22 January 2009
Grammar Stuff
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Table Stuff
\(F\!\) | \(F\!\) | \(F()\!\) | \(F\!\) |
\(\underline{0}\) | \(F_0^{(0)}\!\) | \(\underline{0}\) | \(\underline{(} ~ \underline{)}\) |
\(\underline{1}\) | \(F_1^{(0)}\!\) | \(\underline{1}\) | \(\underline{((} ~ \underline{))}\) |
\(F\!\) | \(F\!\) | \(F(x)\!\) | \(F\!\) | |
\(F(\underline{1})\) | \(F(\underline{0})\) | |||
\(F_0^{(1)}\!\) | \(F_{00}^{(1)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{(} ~ \underline{)}\) |
\(F_1^{(1)}\!\) | \(F_{01}^{(1)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{(} x \underline{)}\) |
\(F_2^{(1)}\!\) | \(F_{10}^{(1)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(x\!\) |
\(F_3^{(1)}\!\) | \(F_{11}^{(1)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{((} ~ \underline{))}\) |
\(F\!\) | \(F\!\) | \(F(x, y)\!\) | \(F\!\) | |||
\(F(\underline{1}, \underline{1})\) | \(F(\underline{1}, \underline{0})\) | \(F(\underline{0}, \underline{1})\) | \(F(\underline{0}, \underline{0})\) | |||
\(F_{0}^{(2)}\!\) | \(F_{0000}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \((~)\) |
\(F_{1}^{(2)}\!\) | \(F_{0001}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \((x)(y)\!\) |
\(F_{2}^{(2)}\!\) | \(F_{0010}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \((x) y\!\) |
\(F_{3}^{(2)}\!\) | \(F_{0011}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \((x)\!\) |
\(F_{4}^{(2)}\!\) | \(F_{0100}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(x (y)\!\) |
\(F_{5}^{(2)}\!\) | \(F_{0101}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \((y)\!\) |
\(F_{6}^{(2)}\!\) | \(F_{0110}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \((x, y)\!\) |
\(F_{7}^{(2)}\!\) | \(F_{0111}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \((x y)\!\) |
\(F_{8}^{(2)}\!\) | \(F_{1000}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \(x y\!\) |
\(F_{9}^{(2)}\!\) | \(F_{1001}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \(((x, y))\!\) |
\(F_{10}^{(2)}\!\) | \(F_{1010}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \(y\!\) |
\(F_{11}^{(2)}\!\) | \(F_{1011}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \((x (y))\!\) |
\(F_{12}^{(2)}\!\) | \(F_{1100}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(x\!\) |
\(F_{13}^{(2)}\!\) | \(F_{1101}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \(((x)y)\!\) |
\(F_{14}^{(2)}\!\) | \(F_{1110}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \(((x)(y))\!\) |
\(F_{15}^{(2)}\!\) | \(F_{1111}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \(((~))\) |
Table 17. Boolean Functions on Two Variables o----------o----------o-------------------------------------------o----------o | Function | Function | F(x, y) | Function | o----------o----------o----------o----------o----------o----------o----------o | | | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% | | o----------o----------o----------o----------o----------o----------o----------o | | | | | | | | | F^2_00 | F^2_0000 | %0% | %0% | %0% | %0% | () | | | | | | | | | | F^2_01 | F^2_0001 | %0% | %0% | %0% | %1% | (x)(y) | | | | | | | | | | F^2_02 | F^2_0010 | %0% | %0% | %1% | %0% | (x) y | | | | | | | | | | F^2_03 | F^2_0011 | %0% | %0% | %1% | %1% | (x) | | | | | | | | | | F^2_04 | F^2_0100 | %0% | %1% | %0% | %0% | x (y) | | | | | | | | | | F^2_05 | F^2_0101 | %0% | %1% | %0% | %1% | (y) | | | | | | | | | | F^2_06 | F^2_0110 | %0% | %1% | %1% | %0% | (x, y) | | | | | | | | | | F^2_07 | F^2_0111 | %0% | %1% | %1% | %1% | (x y) | | | | | | | | | | F^2_08 | F^2_1000 | %1% | %0% | %0% | %0% | x y | | | | | | | | | | F^2_09 | F^2_1001 | %1% | %0% | %0% | %1% | ((x, y)) | | | | | | | | | | F^2_10 | F^2_1010 | %1% | %0% | %1% | %0% | y | | | | | | | | | | F^2_11 | F^2_1011 | %1% | %0% | %1% | %1% | (x (y)) | | | | | | | | | | F^2_12 | F^2_1100 | %1% | %1% | %0% | %0% | x | | | | | | | | | | F^2_13 | F^2_1101 | %1% | %1% | %0% | %1% | ((x) y) | | | | | | | | | | F^2_14 | F^2_1110 | %1% | %1% | %1% | %0% | ((x)(y)) | | | | | | | | | | F^2_15 | F^2_1111 | %1% | %1% | %1% | %1% | (()) | | | | | | | | | o----------o----------o----------o----------o----------o----------o----------o
\(\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}\) |
\(\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}\) |
\(\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}\) |
\(\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}\) |
\(\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}\) |
\(\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}\) |
\(~\!\) | \(x\colon\!\) | \(1~1~0~0\!\) | \(~\!\) | \(~\!\) | \(~\!\) |
\(~\!\) | \(y\colon\!\) | \(1~0~1~0\!\) | \(~\!\) | \(~\!\) | \(~\!\) |
\(f_{0}\!\) | \(f_{0000}\!\) | \(0~0~0~0\!\) | \((~)\!\) | \(\mbox{false}\!\) | \(0\!\) |
\(f_{1}\!\) | \(f_{0001}\!\) | \(0~0~0~1\!\) | \((x)(y)\!\) | \(\mbox{neither}\ x\ \mbox{nor}\ y\!\) | \(\lnot x \land \lnot y\!\) |
\(f_{2}\!\) | \(f_{0010}\!\) | \(0~0~1~0\!\) | \((x)\ y\!\) | \(y\ \mbox{without}\ x\!\) | \(\lnot x \land y\!\) |
\(f_{3}\!\) | \(f_{0011}\!\) | \(0~0~1~1\!\) | \((x)\!\) | \(\mbox{not}\ x\!\) | \(\lnot x\!\) |
\(f_{4}\!\) | \(f_{0100}\!\) | \(0~1~0~0\!\) | \(x\ (y)\!\) | \(x\ \mbox{without}\ y\!\) | \(x \land \lnot y\!\) |
\(f_{5}\!\) | \(f_{0101}\!\) | \(0~1~0~1\!\) | \((y)\!\) | \(\mbox{not}\ y\!\) | \(\lnot y\!\) |
\(f_{6}\!\) | \(f_{0110}\!\) | \(0~1~1~0\!\) | \((x, y)\!\) | \(x\ \mbox{not equal to}\ y\!\) | \(x \ne y\!\) |
\(f_{7}\!\) | \(f_{0111}\!\) | \(0~1~1~1\!\) | \((x\ y)\!\) | \(\mbox{not both}\ x\ \mbox{and}\ y\!\) | \(\lnot x \lor \lnot y\!\) |
\(f_{8}\!\) | \(f_{1000}\!\) | \(1~0~0~0\!\) | \(x\ y\!\) | \(x\ \mbox{and}\ y\!\) | \(x \land y\!\) |
\(f_{9}\!\) | \(f_{1001}\!\) | \(1~0~0~1\!\) | \(((x, y))\!\) | \(x\ \mbox{equal to}\ y\!\) | \(x = y\!\) |
\(f_{10}\!\) | \(f_{1010}\!\) | \(1~0~1~0\!\) | \(y\!\) | \(y\!\) | \(y\!\) |
\(f_{11}\!\) | \(f_{1011}\!\) | \(1~0~1~1\!\) | \((x\ (y))\!\) | \(\mbox{not}\ x\ \mbox{without}\ y\!\) | \(x \Rightarrow y\!\) |
\(f_{12}\!\) | \(f_{1100}\!\) | \(1~1~0~0\!\) | \(x\!\) | \(x\!\) | \(x\!\) |
\(f_{13}\!\) | \(f_{1101}\!\) | \(1~1~0~1\!\) | \(((x)\ y)\!\) | \(\mbox{not}\ y\ \mbox{without}\ x\!\) | \(x \Leftarrow y\!\) |
\(f_{14}\!\) | \(f_{1110}\!\) | \(1~1~1~0\!\) | \(((x)(y))\!\) | \(x\ \mbox{or}\ y\!\) | \(x \lor y\!\) |
\(f_{15}\!\) | \(f_{1111}\!\) | \(1~1~1~1\!\) | \(((~))\!\) | \(\mbox{true}\!\) | \(1\!\) |
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