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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
differential propositional calculus : appendix 1
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(Application)
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Note. The following Tables are best viewed in the Page Image mode.
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 |
| $\mathcal{L}_1$ |
$\mathcal{L}_2$ |
|
$\mathcal{L}_3$ |
$\mathcal{L}_4$ |
$\mathcal{L}_5$ |
$\mathcal{L}_6$ |
| |
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$x =$ |
1 1 0 0 |
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$y =$ |
1 0 1 0 |
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| $f_{0}$ |
$f_{0000}$ |
|
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 |
$(x,\ y)$ |
$x\ \operatorname{not~equal~to}\ y$ |
$x \ne y$ |
| $f_{7}$ |
$f_{0111}$ |
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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 |
$y$ |
$y$ |
$y$ |
| $f_{11}$ |
$f_{1011}$ |
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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 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 |
| $\mathcal{L}_1$ |
$\mathcal{L}_2$ |
|
$\mathcal{L}_3$ |
$\mathcal{L}_4$ |
$\mathcal{L}_5$ |
$\mathcal{L}_6$ |
| |
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$x =$ |
1 1 0 0 |
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| |
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$y =$ |
1 0 1 0 |
|
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| $f_{0}$ |
$f_{0000}$ |
|
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 |
$(x,\ y)$ |
$x\ \operatorname{not~equal~to}\ y$ |
$x \ne y$ |
| $f_{9}$ |
$f_{1001}$ |
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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 |
$y$ |
$y$ |
$y$ |
| $f_{7}$ |
$f_{0111}$ |
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0 1 1 1 |
$(x\ y)$ |
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ |
$\lnot x \lor \lnot y$ |
| $f_{11}$ |
$f_{1011}$ |
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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 $\{ \operatorname{d}x, \operatorname{d}y \}$ |
| |
|
$\operatorname{T}_{11}$ |
$\operatorname{T}_{10}$ |
$\operatorname{T}_{01}$ |
$\operatorname{T}_{00}$ |
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$f$ |
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ |
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ |
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ |
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ |
| $f_{0}$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
| $f_{1}$ |
$(x)(y)$ |
$x\ y$ |
$x\ (y)$ |
$(x)\ y$ |
$(x)(y)$ |
| $f_{2}$ |
$(x)\ y$ |
$x\ (y)$ |
$x\ y$ |
$(x)(y)$ |
$(x)\ y$ |
| $f_{4}$ |
$x\ (y)$ |
$(x)\ y$ |
$(x)(y)$ |
$x\ y$ |
$x\ (y)$ |
| $f_{8}$ |
$x\ y$ |
$(x)(y)$ |
$(x)\ y$ |
$x\ (y)$ |
$x\ y$ |
| $f_{3}$ |
$(x)$ |
$x$ |
$x$ |
$(x)$ |
$(x)$ |
| $f_{12}$ |
$x$ |
$(x)$ |
$(x)$ |
$x$ |
$x$ |
| $f_{6}$ |
$(x,\ y)$ |
$(x,\ y)$ |
$((x,\ y))$ |
$((x,\ y))$ |
$(x,\ y)$ |
| $f_{9}$ |
$((x,\ y))$ |
$((x,\ y))$ |
$(x,\ y)$ |
$(x,\ y)$ |
$((x,\ y))$ |
| $f_{5}$ |
$(y)$ |
$y$ |
$(y)$ |
$y$ |
$(y)$ |
| $f_{10}$ |
$y$ |
$(y)$ |
$y$ |
$(y)$ |
$y$ |
| $f_{7}$ |
$(x\ y)$ |
$((x)(y))$ |
$((x)\ y)$ |
$(x\ (y))$ |
$(x\ y)$ |
| $f_{11}$ |
$(x\ (y))$ |
$((x)\ y)$ |
$((x)(y))$ |
$(x\ y)$ |
$(x\ (y))$ |
| $f_{13}$ |
$((x)\ y)$ |
$(x\ (y))$ |
$(x\ y)$ |
$((x)(y))$ |
$((x)\ y)$ |
| $f_{14}$ |
$((x)(y))$ |
$(x\ y)$ |
$(x\ (y))$ |
$((x)\ y)$ |
$((x)(y))$ |
| $f_{15}$ |
$((~))$ |
$((~))$ |
$((~))$ |
$((~))$ |
$((~))$ |
| Fixed Point Total: |
4 |
4 |
4 |
16 |
| Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$ |
| |
$f$ |
$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ |
$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ |
$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ |
$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ |
| $f_{0}$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
| $f_{1}$ |
$(x)(y)$ |
$((x,\ y))$ |
$(y)$ |
$(x)$ |
$(~)$ |
| $f_{2}$ |
$(x)\ y$ |
$(x,\ y)$ |
$y$ |
$(x)$ |
$(~)$ |
| $f_{4}$ |
$x\ (y)$ |
$(x,\ y)$ |
$(y)$ |
$x$ |
$(~)$ |
| $f_{8}$ |
$x\ y$ |
$((x,\ y))$ |
$y$ |
$x$ |
$(~)$ |
| $f_{3}$ |
$(x)$ |
$((~))$ |
$((~))$ |
$(~)$ |
$(~)$ |
| $f_{12}$ |
$x$ |
$((~))$ |
$((~))$ |
$(~)$ |
$(~)$ |
| $f_{6}$ |
$(x,\ y)$ |
$(~)$ |
$((~))$ |
$((~))$ |
$(~)$ |
| $f_{9}$ |
$((x,\ y))$ |
$(~)$ |
$((~))$ |
$((~))$ |
$(~)$ |
| $f_{5}$ |
$(y)$ |
$((~))$ |
$(~)$ |
$((~))$ |
$(~)$ |
| $f_{10}$ |
$y$ |
$((~))$ |
$(~)$ |
$((~))$ |
$(~)$ |
| $f_{7}$ |
$(x\ y)$ |
$((x,\ y))$ |
$y$ |
$x$ |
$(~)$ |
| $f_{11}$ |
$(x\ (y))$ |
$(x,\ y)$ |
$(y)$ |
$x$ |
$(~)$ |
| $f_{13}$ |
$((x)\ y)$ |
$(x,\ y)$ |
$y$ |
$(x)$ |
$(~)$ |
| $f_{14}$ |
$((x)(y))$ |
$((x,\ y))$ |
$(y)$ |
$(x)$ |
$(~)$ |
| $f_{15}$ |
$((~))$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
| Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$ |
| |
$f$ |
$\operatorname{E}f|_{x\ y}$ |
$\operatorname{E}f|_{x (y)}$ |
$\operatorname{E}f|_{(x) y}$ |
$\operatorname{E}f|_{(x)(y)}$ |
| $f_{0}$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
| $f_{1}$ |
$(x)(y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$(\operatorname{d}x)(\operatorname{d}y)$ |
| $f_{2}$ |
$(x)\ y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$(\operatorname{d}x)(\operatorname{d}y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
| $f_{4}$ |
$x\ (y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$(\operatorname{d}x)(\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
| $f_{8}$ |
$x\ y$ |
$(\operatorname{d}x)(\operatorname{d}y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
| $f_{3}$ |
$(x)$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
$(\operatorname{d}x)$ |
$(\operatorname{d}x)$ |
| $f_{12}$ |
$x$ |
$(\operatorname{d}x)$ |
$(\operatorname{d}x)$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
| $f_{6}$ |
$(x,\ y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$((\operatorname{d}x,\ \operatorname{d}y))$ |
$((\operatorname{d}x,\ \operatorname{d}y))$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
| $f_{9}$ |
$((x,\ y))$ |
$((\operatorname{d}x,\ \operatorname{d}y))$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$((\operatorname{d}x,\ \operatorname{d}y))$ |
| $f_{5}$ |
$(y)$ |
$\operatorname{d}y$ |
$(\operatorname{d}y)$ |
$\operatorname{d}y$ |
$(\operatorname{d}y)$ |
| $f_{10}$ |
$y$ |
$(\operatorname{d}y)$ |
$\operatorname{d}y$ |
$(\operatorname{d}y)$ |
$\operatorname{d}y$ |
| $f_{7}$ |
$(x\ y)$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$((\operatorname{d}x)\ \operatorname{d}y)$ |
$(\operatorname{d}x\ (\operatorname{d}y))$ |
$(\operatorname{d}x\ \operatorname{d}y)$ |
| $f_{11}$ |
$(x\ (y))$ |
$((\operatorname{d}x)\ \operatorname{d}y)$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$(\operatorname{d}x\ \operatorname{d}y)$ |
$(\operatorname{d}x\ (\operatorname{d}y))$ |
| $f_{13}$ |
$((x)\ y)$ |
$(\operatorname{d}x\ (\operatorname{d}y))$ |
$(\operatorname{d}x\ \operatorname{d}y)$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$((\operatorname{d}x)\ \operatorname{d}y)$ |
| $f_{14}$ |
$((x)(y))$ |
$(\operatorname{d}x\ \operatorname{d}y)$ |
$(\operatorname{d}x\ (\operatorname{d}y))$ |
$((\operatorname{d}x)\ \operatorname{d}y)$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
| $f_{15}$ |
$((~))$ |
$((~))$ |
$((~))$ |
$((~))$ |
$((~))$ |
| Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$ |
| |
$f$ |
$\operatorname{D}f|_{x\ y}$ |
$\operatorname{D}f|_{x (y)}$ |
$\operatorname{D}f|_{(x) y}$ |
$\operatorname{D}f|_{(x)(y)}$ |
| $f_{0}$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
| $f_{1}$ |
$(x)(y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
| $f_{2}$ |
$(x)\ y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
| $f_{4}$ |
$x\ (y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
| $f_{8}$ |
$x\ y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
| $f_{3}$ |
$(x)$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
| $f_{12}$ |
$x$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
$\operatorname{d}x$ |
| $f_{6}$ |
$(x,\ y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
| $f_{9}$ |
$((x,\ y))$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
$(\operatorname{d}x,\ \operatorname{d}y)$ |
| $f_{5}$ |
$(y)$ |
$\operatorname{d}y$ |
$\operatorname{d}y$ |
$\operatorname{d}y$ |
$\operatorname{d}y$ |
| $f_{10}$ |
$y$ |
$\operatorname{d}y$ |
$\operatorname{d}y$ |
$\operatorname{d}y$ |
$\operatorname{d}y$ |
| $f_{7}$ |
$(x\ y)$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
| $f_{11}$ |
$(x\ (y))$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
| $f_{13}$ |
$((x)\ y)$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
| $f_{14}$ |
$((x)(y))$ |
$\operatorname{d}x\ \operatorname{d}y$ |
$\operatorname{d}x\ (\operatorname{d}y)$ |
$(\operatorname{d}x)\ \operatorname{d}y$ |
$((\operatorname{d}x)(\operatorname{d}y))$ |
| $f_{15}$ |
$((~))$ |
$(~)$ |
$(~)$ |
$(~)$ |
$(~)$ |
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"differential propositional calculus : appendix 1" is owned by Jon Awbrey.
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(view preamble | get metadata)
Cross-references: similarity, variables, Boolean functions, expressions, equivalent
There is 1 reference to this entry.
This is version 16 of differential propositional calculus : appendix 1, born on 2008-06-06, modified 2008-08-03.
Object id is 10677, canonical name is DifferentialPropositionalCalculusAppendices.
Accessed 1016 times total.
Classification:
| AMS MSC: | 03B05 (Mathematical logic and foundations :: General logic :: Classical propositional logic) | | | 03B42 (Mathematical logic and foundations :: General logic :: Logic of knowledge and belief) | | | 03B44 (Mathematical logic and foundations :: General logic :: Temporal logic) | | | 34G99 (Ordinary differential equations :: Differential equations in abstract spaces :: Miscellaneous) | | | 39A12 (Difference and functional equations :: Difference equations :: Discrete version of topics in analysis) | | | 53A40 (Differential geometry :: Classical differential geometry :: Other special differential geometries) |
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Pending Errata and Addenda
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