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[parent] differential propositional calculus : appendix 1 (Application)

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Contents

Table A1. Propositional Forms on Two Variables

Table A1 lists equivalent expressions for the boolean functions of two variables in a number of different notational systems.

Table A1. Propositional Forms on Two Variables
$\mathcal{L}_1$ $\mathcal{L}_2$   $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$
    $x =$ 1 1 0 0      
    $y =$ 1 0 1 0      
$f_{0}$ $f_{0000}$   0 0 0 0 $(~)$ $\operatorname{false}$ 0
$f_{1}$ $f_{0001}$   0 0 0 1 $(x)(y)$ $\operatorname{neither}\ x\ \operatorname{nor}\ y$ $\lnot x \land \lnot y$
$f_{2}$ $f_{0010}$   0 0 1 0 $(x)\ y$ $y\ \operatorname{without}\ x$ $\lnot x \land y$
$f_{3}$ $f_{0011}$   0 0 1 1 $(x)$ $\operatorname{not}\ x$ $\lnot x$
$f_{4}$ $f_{0100}$   0 1 0 0 $x\ (y)$ $x\ \operatorname{without}\ y$ $x \land \lnot y$
$f_{5}$ $f_{0101}$   0 1 0 1 $(y)$ $\operatorname{not}\ y$ $\lnot y$
$f_{6}$ $f_{0110}$   0 1 1 0 $(x,\ y)$ $x\ \operatorname{not~equal~to}\ y$ $x \ne y$
$f_{7}$ $f_{0111}$   0 1 1 1 $(x\ y)$ $\operatorname{not~both}\ x\ \operatorname{and}\ y$ $\lnot x \lor \lnot y$
$f_{8}$ $f_{1000}$   1 0 0 0 $x\ y$ $x\ \operatorname{and}\ y$ $x \land y$
$f_{9}$ $f_{1001}$   1 0 0 1 $((x,\ y))$ $x\ \operatorname{equal~to}\ y$ $x = y$
$f_{10}$ $f_{1010}$   1 0 1 0 $y$ $y$ $y$
$f_{11}$ $f_{1011}$   1 0 1 1 $(x\ (y))$ $\operatorname{not}\ x\ \operatorname{without}\ y$ $x \Rightarrow y$
$f_{12}$ $f_{1100}$   1 1 0 0 $x$ $x$ $x$
$f_{13}$ $f_{1101}$   1 1 0 1 $((x)\ y)$ $\operatorname{not}\ y\ \operatorname{without}\ x$ $x \Leftarrow y$
$f_{14}$ $f_{1110}$   1 1 1 0 $((x)(y))$ $x\ \operatorname{or}\ y$ $x \lor y$
$f_{15}$ $f_{1111}$   1 1 1 1 $((~))$ $\operatorname{true}$ $1$

Table A2. Propositional Forms on Two Variables

Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.

Table A2. Propositional Forms on Two Variables
$\mathcal{L}_1$ $\mathcal{L}_2$   $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$
    $x =$ 1 1 0 0      
    $y =$ 1 0 1 0      
$f_{0}$ $f_{0000}$   0 0 0 0 $(~)$ $\operatorname{false}$ 0
$f_{1}$ $f_{0001}$   0 0 0 1 $(x)(y)$ $\operatorname{neither}\ x\ \operatorname{nor}\ y$ $\lnot x \land \lnot y$
$f_{2}$ $f_{0010}$   0 0 1 0 $(x)\ y$ $y\ \operatorname{without}\ x$ $\lnot x \land y$
$f_{4}$ $f_{0100}$   0 1 0 0 $x\ (y)$ $x\ \operatorname{without}\ y$ $x \land \lnot y$
$f_{8}$ $f_{1000}$   1 0 0 0 $x\ y$ $x\ \operatorname{and}\ y$ $x \land y$
$f_{3}$ $f_{0011}$   0 0 1 1 $(x)$ $\operatorname{not}\ x$ $\lnot x$
$f_{12}$ $f_{1100}$   1 1 0 0 $x$ $x$ $x$
$f_{6}$ $f_{0110}$   0 1 1 0 $(x,\ y)$ $x\ \operatorname{not~equal~to}\ y$ $x \ne y$
$f_{9}$ $f_{1001}$   1 0 0 1 $((x,\ y))$ $x\ \operatorname{equal~to}\ y$ $x = y$
$f_{5}$ $f_{0101}$   0 1 0 1 $(y)$ $\operatorname{not}\ y$ $\lnot y$
$f_{10}$ $f_{1010}$   1 0 1 0 $y$ $y$ $y$
$f_{7}$ $f_{0111}$   0 1 1 1 $(x\ y)$ $\operatorname{not~both}\ x\ \operatorname{and}\ y$ $\lnot x \lor \lnot y$
$f_{11}$ $f_{1011}$   1 0 1 1 $(x\ (y))$ $\operatorname{not}\ x\ \operatorname{without}\ y$ $x \Rightarrow y$
$f_{13}$ $f_{1101}$   1 1 0 1 $((x)\ y)$ $\operatorname{not}\ y\ \operatorname{without}\ x$ $x \Leftarrow y$
$f_{14}$ $f_{1110}$   1 1 1 0 $((x)(y))$ $x\ \operatorname{or}\ y$ $x \lor y$
$f_{15}$ $f_{1111}$   1 1 1 1 $((~))$ $\operatorname{true}$ $1$

Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$

Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$
    $\operatorname{T}_{11}$ $\operatorname{T}_{10}$ $\operatorname{T}_{01}$ $\operatorname{T}_{00}$
  $f$ $\operatorname{E}f\vert _{\operatorname{d}x\ \operatorname{d}y}$ $\operatorname{E}f\vert _{\operatorname{d}x (\operatorname{d}y)}$ $\operatorname{E}f\vert _{(\operatorname{d}x) \operatorname{d}y}$ $\operatorname{E}f\vert _{(\operatorname{d}x)(\operatorname{d}y)}$
$f_{0}$ $(~)$ $(~)$ $(~)$ $(~)$ $(~)$
$f_{1}$ $(x)(y)$ $x\ y$ $x\ (y)$ $(x)\ y$ $(x)(y)$
$f_{2}$ $(x)\ y$ $x\ (y)$ $x\ y$ $(x)(y)$ $(x)\ y$
$f_{4}$ $x\ (y)$ $(x)\ y$ $(x)(y)$ $x\ y$ $x\ (y)$
$f_{8}$ $x\ y$ $(x)(y)$ $(x)\ y$ $x\ (y)$ $x\ y$
$f_{3}$ $(x)$ $x$ $x$ $(x)$ $(x)$
$f_{12}$ $x$ $(x)$ $(x)$ $x$ $x$
$f_{6}$ $(x,\ y)$ $(x,\ y)$ $((x,\ y))$ $((x,\ y))$ $(x,\ y)$
$f_{9}$ $((x,\ y))$ $((x,\ y))$ $(x,\ y)$ $(x,\ y)$ $((x,\ y))$
$f_{5}$ $(y)$ $y$ $(y)$ $y$ $(y)$
$f_{10}$ $y$ $(y)$ $y$ $(y)$ $y$
$f_{7}$ $(x\ y)$ $((x)(y))$ $((x)\ y)$ $(x\ (y))$ $(x\ y)$
$f_{11}$ $(x\ (y))$ $((x)\ y)$ $((x)(y))$ $(x\ y)$ $(x\ (y))$
$f_{13}$ $((x)\ y)$ $(x\ (y))$ $(x\ y)$ $((x)(y))$ $((x)\ y)$
$f_{14}$ $((x)(y))$ $(x\ y)$ $(x\ (y))$ $((x)\ y)$ $((x)(y))$
$f_{15}$ $((~))$ $((~))$ $((~))$ $((~))$ $((~))$
Fixed Point Total: 4 4 4 16

Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$

Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$
  $f$ $\operatorname{D}f\vert _{\operatorname{d}x\ \operatorname{d}y}$ $\operatorname{D}f\vert _{\operatorname{d}x (\operatorname{d}y)}$ $\operatorname{D}f\vert _{(\operatorname{d}x) \operatorname{d}y}$ $\operatorname{D}f\vert _{(\operatorname{d}x)(\operatorname{d}y)}$
$f_{0}$ $(~)$ $(~)$ $(~)$ $(~)$ $(~)$
$f_{1}$ $(x)(y)$ $((x,\ y))$ $(y)$ $(x)$ $(~)$
$f_{2}$ $(x)\ y$ $(x,\ y)$ $y$ $(x)$ $(~)$
$f_{4}$ $x\ (y)$ $(x,\ y)$ $(y)$ $x$ $(~)$
$f_{8}$ $x\ y$ $((x,\ y))$ $y$ $x$ $(~)$
$f_{3}$ $(x)$ $((~))$ $((~))$ $(~)$ $(~)$
$f_{12}$ $x$ $((~))$ $((~))$ $(~)$ $(~)$
$f_{6}$ $(x,\ y)$ $(~)$ $((~))$ $((~))$ $(~)$
$f_{9}$ $((x,\ y))$ $(~)$ $((~))$ $((~))$ $(~)$
$f_{5}$ $(y)$ $((~))$ $(~)$ $((~))$ $(~)$
$f_{10}$ $y$ $((~))$ $(~)$ $((~))$ $(~)$
$f_{7}$ $(x\ y)$ $((x,\ y))$ $y$ $x$ $(~)$
$f_{11}$ $(x\ (y))$ $(x,\ y)$ $(y)$ $x$ $(~)$
$f_{13}$ $((x)\ y)$ $(x,\ y)$ $y$ $(x)$ $(~)$
$f_{14}$ $((x)(y))$ $((x,\ y))$ $(y)$ $(x)$ $(~)$
$f_{15}$ $((~))$ $(~)$ $(~)$ $(~)$ $(~)$

Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$

Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$
  $f$ $\operatorname{E}f\vert _{x\ y}$ $\operatorname{E}f\vert _{x (y)}$ $\operatorname{E}f\vert _{(x) y}$ $\operatorname{E}f\vert _{(x)(y)}$
$f_{0}$ $(~)$ $(~)$ $(~)$ $(~)$ $(~)$
$f_{1}$ $(x)(y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $(\operatorname{d}x)(\operatorname{d}y)$
$f_{2}$ $(x)\ y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $(\operatorname{d}x)(\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$
$f_{4}$ $x\ (y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $(\operatorname{d}x)(\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
$f_{8}$ $x\ y$ $(\operatorname{d}x)(\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
$f_{3}$ $(x)$ $\operatorname{d}x$ $\operatorname{d}x$ $(\operatorname{d}x)$ $(\operatorname{d}x)$
$f_{12}$ $x$ $(\operatorname{d}x)$ $(\operatorname{d}x)$ $\operatorname{d}x$ $\operatorname{d}x$
$f_{6}$ $(x,\ y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $((\operatorname{d}x,\ \operatorname{d}y))$ $((\operatorname{d}x,\ \operatorname{d}y))$ $(\operatorname{d}x,\ \operatorname{d}y)$
$f_{9}$ $((x,\ y))$ $((\operatorname{d}x,\ \operatorname{d}y))$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $((\operatorname{d}x,\ \operatorname{d}y))$
$f_{5}$ $(y)$ $\operatorname{d}y$ $(\operatorname{d}y)$ $\operatorname{d}y$ $(\operatorname{d}y)$
$f_{10}$ $y$ $(\operatorname{d}y)$ $\operatorname{d}y$ $(\operatorname{d}y)$ $\operatorname{d}y$
$f_{7}$ $(x\ y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$
$f_{11}$ $(x\ (y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$
$f_{13}$ $((x)\ y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$
$f_{14}$ $((x)(y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{15}$ $((~))$ $((~))$ $((~))$ $((~))$ $((~))$

Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$

Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$
  $f$ $\operatorname{D}f\vert _{x\ y}$ $\operatorname{D}f\vert _{x (y)}$ $\operatorname{D}f\vert _{(x) y}$ $\operatorname{D}f\vert _{(x)(y)}$
$f_{0}$ $(~)$ $(~)$ $(~)$ $(~)$ $(~)$
$f_{1}$ $(x)(y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{2}$ $(x)\ y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$
$f_{4}$ $x\ (y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
$f_{8}$ $x\ y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
$f_{3}$ $(x)$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$
$f_{12}$ $x$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$
$f_{6}$ $(x,\ y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
$f_{9}$ $((x,\ y))$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
$f_{5}$ $(y)$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$
$f_{10}$ $y$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$
$f_{7}$ $(x\ y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
$f_{11}$ $(x\ (y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
$f_{13}$ $((x)\ y)$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$
$f_{14}$ $((x)(y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{15}$ $((~))$ $(~)$ $(~)$ $(~)$ $(~)$



"differential propositional calculus : appendix 1" is owned by Jon Awbrey.

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See Also: differential logic, minimal negation operator


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Cross-references: boolean functions

This is version 1 of differential propositional calculus : appendix 1, born on 2009-05-25.
Object id is 778, canonical name is DifferentialPropositionalCalculusAppendix1.
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Physics Classification: 02. (Mathematical methods in physics)
 02.10.Ab (Logic and set theory)
 02.10.Ox (Combinatorics; graph theory)
 02.10.Ud (Linear algebra)
 02.20.-a (Group theory )
 02.30.-f (Function theory, analysis)
 02.40.-k (Geometry, differential geometry, and topology )
 02.40.Yy (Geometric mechanics )
 02.50.Tt (Inference methods)
 02.70.-c (Computational techniques )
 02.70.Bf (Finite-difference methods)
 02.70.Wz (Symbolic computation )
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