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| {{DISPLAYTITLE:Differential Propositional Calculus}} | | {{DISPLAYTITLE:Differential Propositional Calculus}} |
| + | |
| A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe. | | A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe. |
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| {| align="center" cellpadding="10" style="text-align:center" | | {| align="center" cellpadding="10" style="text-align:center" |
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− | <p>[[Image:Differential_Propositional_Calculus_2.jpg|500px]]</p> | + | <p>[[Differential_Propositional_Calculus_2.jpg|500px]]</p> |
| <p>'''Figure 2. Same Names, Different Habitations'''</p> | | <p>'''Figure 2. Same Names, Different Habitations'''</p> |
| |} | | |} |
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| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | <math>((x),(y),(z))\!</math> | + | | <math>((x),(y),(z))~\!</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
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| | | | | |
| <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br> | | <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br> |
− | <math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br> | + | <math>(\mathbb{B}^n\ +\!\to \mathbb{B})\!</math><br> |
| <math>[\mathbb{B}^n]</math> | | <math>[\mathbb{B}^n]</math> |
| |} | | |} |
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| </ul> | | </ul> |
| | | |
− | In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>n = 3,\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)} \texttt{(} a_2 \texttt{)} \texttt{(} a_3 \texttt{)}.\!</math> | + | In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>n = 3,~\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)} \texttt{(} a_2 \texttt{)} \texttt{(} a_3 \texttt{)}.\!</math> |
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| The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}\!</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. | | The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}\!</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. |
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| An initial universe of discourse, <math>A^\circ,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\operatorname{E}A^\circ.</math> The construction of <math>\operatorname{E}A^\circ</math> can be described in the following stages: | | An initial universe of discourse, <math>A^\circ,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\operatorname{E}A^\circ.</math> The construction of <math>\operatorname{E}A^\circ</math> can be described in the following stages: |
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− | :* <p>The initial alphabet, <math>\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace,\!</math> is extended by a ''first order differential alphabet'', <math>\operatorname{d}\mathfrak{A} = \lbrace\!</math> “<math>\operatorname{d}a_1\!</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n\!</math>” <math>\rbrace,\!</math> resulting in a ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:</p><blockquote><math>\operatorname{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>”<math>,\!</math> “<math>\operatorname{d}a_1\!</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n\!</math>” <math>\rbrace.\!</math></blockquote>
| + | <ul> |
| + | |
| + | <li> |
| + | <p>The initial alphabet, <math>\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \},\!</math> is extended by a ''first order differential alphabet'', <math>\operatorname{d}\mathfrak{A} = \{ {}^{\backprime\backprime} \operatorname{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \operatorname{d}a_n {}^{\prime\prime} \},\!</math> resulting in a ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:</p> |
| + | |
| + | {| align="center" cellspacing="8" width="90%" |
| + | | |
| + | <math>\operatorname{E}\mathfrak{A} ~=~ \mathfrak{A} ~\cup~ \operatorname{d}\mathfrak{A} ~=~ \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime}, {}^{\backprime\backprime} \operatorname{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \operatorname{d}a_n {}^{\prime\prime} \}.\!</math> |
| + | |} |
| + | </li> |
| + | |
| + | <li> |
| + | <p>The initial basis, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},\!</math> is extended by a ''first order differential basis'', <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},\!</math> resulting in a ''first order extended basis'', <math>\operatorname{E}\mathcal{A},\!</math> defined as follows:</p> |
| + | |
| + | {| align="center" cellspacing="8" width="90%" |
| + | | |
| + | <math>\operatorname{E}\mathcal{A} ~=~ \mathcal{A} ~\cup~ \operatorname{d}\mathcal{A} ~=~ \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.\!</math> |
| + | |} |
| + | </li> |
| + | |
| + | <li> |
| + | <p>The initial space, <math>A = \langle a_1, \ldots, a_n \rangle,\!</math> is extended by a ''first order differential space'' or ''tangent space'', <math>\operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,\!</math> at each point of <math>A,\!</math> resulting in a ''first order extended space'' or ''tangent bundle space'', <math>\operatorname{E}A,\!</math> defined as follows:</p> |
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− | :* <p>The initial basis, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> is extended by a ''first order differential basis'', <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},</math> resulting in a ''first order extended basis'', <math>\operatorname{E}\mathcal{A},</math> defined as follows:</p><blockquote><math>\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math></blockquote>
| + | {| align="center" cellspacing="8" width="90%" |
| + | | |
| + | <math>\operatorname{E}A ~=~ A ~\times~ \operatorname{d}A ~=~ \langle \operatorname{E}\mathcal{A} \rangle ~=~ \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle ~=~ \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.\!</math> |
| + | |} |
| + | </li> |
| | | |
− | :* <p>The initial space, <math>A = \langle a_1, \ldots, a_n \rangle,</math> is extended by a ''first order differential space'' or ''tangent space'', <math>\operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math> at each point of <math>A,\!</math> resulting in a ''first order extended space'' or ''tangent bundle space'', <math>\operatorname{E}A,</math> defined as follows:</p><blockquote><math>\operatorname{E}A = A \times \operatorname{d}A = \langle \operatorname{E}\mathcal{A} \rangle = \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math></blockquote>
| + | <li> |
| + | <p>Finally, the initial universe, <math>A^\circ = [ a_1, \ldots, a_n ],\!</math> is extended by a ''first order differential universe'' or ''tangent universe'', <math>\operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ],\!</math> at each point of <math>A^\circ,\!</math> resulting in a ''first order extended universe'' or ''tangent bundle universe'', <math>\operatorname{E}A^\circ,\!</math> defined as follows:</p> |
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− | :* <p>Finally, the initial universe, <math>A^\circ = [ a_1, \ldots, a_n ],</math> is extended by a ''first order differential universe'' or ''tangent universe'', <math>\operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ],</math> at each point of <math>A^\circ,</math> resulting in a ''first order extended universe'' or ''tangent bundle universe'', <math>\operatorname{E}A^\circ,</math> defined as follows:</p><blockquote><math>\operatorname{E}A^\circ = [ \operatorname{E}\mathcal{A} ] = [ \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} ] = [ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n ].</math></blockquote><p>This gives <math>\operatorname{E}A^\circ</math> the type:</p><blockquote><math>[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).</math></blockquote>
| + | {| align="center" cellspacing="8" width="90%" |
| + | | |
| + | <math>\operatorname{E}A^\circ ~=~ [ \operatorname{E}\mathcal{A} ] ~=~ [ \mathcal{A} ~\cup~ \operatorname{d}\mathcal{A} ] ~=~ [ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n ].\!</math> |
| + | |} |
| + | |
| + | <p>This gives <math>\operatorname{E}A^\circ\!</math> the type:</p> |
| + | |
| + | {| align="center" cellspacing="8" width="90%" |
| + | | |
| + | <math>[ \mathbb{B}^n \times \mathbb{D}^n ] ~=~ (\mathbb{B}^n \times \mathbb{D}^n\ +\!\!\to \mathbb{B}) ~=~ (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).\!</math> |
| + | |} |
| + | </li> |
| + | |
| + | </ul> |
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| A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]]. | | A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]]. |
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| | <math>\mathbb{D}^n \to \mathbb{B}</math> | | | <math>\mathbb{D}^n \to \mathbb{B}</math> |
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− | | <math>\operatorname{d}A^\circ</math> | + | | <math>\operatorname{d}A^\circ\!</math> |
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− | <math>[\operatorname{d}\mathcal{A}]</math><br> | + | <math>[\operatorname{d}\mathcal{A}]\!</math><br> |
− | <math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br> | + | <math>(\operatorname{d}A, \operatorname{d}A^\uparrow)\!</math><br> |
− | <math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br> | + | <math>(\operatorname{d}A\ +\!\to \mathbb{B})\!</math><br> |
− | <math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br> | + | <math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))\!</math><br> |
− | <math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math> | + | <math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]\!</math> |
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| Tangent universe<br> | | Tangent universe<br> |
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| | height="36px" | <p><math>f_{0000}\!</math></p> | | | height="36px" | <p><math>f_{0000}\!</math></p> |
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− | | height="36px" | <p><math>f_{0001}\!</math></p> | + | | height="36px" | <p><math>f_{0001}~\!</math></p> |
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| | height="36px" | <p><math>f_{0010}\!</math></p> | | | height="36px" | <p><math>f_{0010}\!</math></p> |
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| | height="36px" | <p><math>(x)(y)\!</math></p> | | | height="36px" | <p><math>(x)(y)\!</math></p> |
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− | | height="36px" | <p><math>(x)\ y\!</math></p> | + | | height="36px" | <p><math>(x)~y\!</math></p> |
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| | height="36px" | <p><math>(x)\!</math></p> | | | height="36px" | <p><math>(x)\!</math></p> |
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− | | height="36px" | <p><math>x\ (y)\!</math></p> | + | | height="36px" | <p><math>x~(y)\!</math></p> |
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| | height="36px" | <p><math>(y)\!</math></p> | | | height="36px" | <p><math>(y)\!</math></p> |
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− | | height="36px" | <p><math>(x,\ y)\!</math></p> | + | | height="36px" | <p><math>(x,~y)\!</math></p> |
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− | | height="36px" | <p><math>(x\ y)\!</math></p> | + | | height="36px" | <p><math>(x~y)\!</math></p> |
| |} | | |} |
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| | height="36px" | <p><math>\operatorname{false}</math></p> | | | height="36px" | <p><math>\operatorname{false}</math></p> |
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− | | height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{neither}~ x ~\operatorname{nor}~ y</math></p> |
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− | | height="36px" | <p><math>y\ \operatorname{without}\ x</math></p> | + | | height="36px" | <p><math>y ~\operatorname{without}~ x</math></p> |
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− | | height="36px" | <p><math>\operatorname{not}\ x</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ x</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{without}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{without}~ y</math></p> |
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− | | height="36px" | <p><math>\operatorname{not}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ y</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{not~equal~to}~ y</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{not~both}~ x ~\operatorname{and}~ y</math></p> |
| |} | | |} |
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| {| align="center" | | {| align="center" |
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− | | height="36px" | <p><math>x\ y\!</math></p> | + | | height="36px" | <p><math>x~y\!</math></p> |
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− | | height="36px" | <p><math>((x,\ y))\!</math></p> | + | | height="36px" | <p><math>((x,~y))\!</math></p> |
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| | height="36px" | <p><math>y\!</math></p> | | | height="36px" | <p><math>y\!</math></p> |
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− | | height="36px" | <p><math>(x\ (y))\!</math></p> | + | | height="36px" | <p><math>(x~(y))\!</math></p> |
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| | height="36px" | <p><math>x\!</math></p> | | | height="36px" | <p><math>x\!</math></p> |
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− | | height="36px" | <p><math>((x)\ y)\!</math></p> | + | | height="36px" | <p><math>((x)~y)\!</math></p> |
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| | height="36px" | <p><math>((x)(y))\!</math></p> | | | height="36px" | <p><math>((x)(y))\!</math></p> |
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| {| align="center" | | {| align="center" |
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− | | height="36px" | <p><math>x\ \operatorname{and}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{and}~ y</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{equal~to}~ y</math></p> |
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| | height="36px" | <p><math>y\!</math></p> | | | height="36px" | <p><math>y\!</math></p> |
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− | | height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ x ~\operatorname{without}~ y</math></p> |
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| | height="36px" | <p><math>x\!</math></p> | | | height="36px" | <p><math>x\!</math></p> |
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− | | height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ y ~\operatorname{without}~ x</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{or}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{or}~ y</math></p> |
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| | height="36px" | <p><math>\operatorname{true}</math></p> | | | height="36px" | <p><math>\operatorname{true}</math></p> |
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| {| align="center" | | {| align="center" |
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− | | height="36px" | <p><math>f_{0001}\!</math></p> | + | | height="36px" | <p><math>f_{0001}~\!</math></p> |
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| | height="36px" | <p><math>f_{0010}\!</math></p> | | | height="36px" | <p><math>f_{0010}\!</math></p> |
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| | height="36px" | <p><math>(x)(y)\!</math></p> | | | height="36px" | <p><math>(x)(y)\!</math></p> |
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− | | height="36px" | <p><math>(x)\ y\!</math></p> | + | | height="36px" | <p><math>(x)~y\!</math></p> |
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− | | height="36px" | <p><math>x\ (y)\!</math></p> | + | | height="36px" | <p><math>x~(y)\!</math></p> |
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− | | height="36px" | <p><math>x\ y\!</math></p> | + | | height="36px" | <p><math>x~y\!</math></p> |
| |} | | |} |
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| {| align="center" | | {| align="center" |
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− | | height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{neither}~ x ~\operatorname{nor}~ y</math></p> |
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− | | height="36px" | <p><math>y\ \operatorname{without}\ x</math></p> | + | | height="36px" | <p><math>y ~\operatorname{without}~ x</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{without}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{without}~ y</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{and}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{and}~ y</math></p> |
| |} | | |} |
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| | height="36px" | <p><math>x\!</math></p> | | | height="36px" | <p><math>x\!</math></p> |
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− | | height="36px" | <p><math>(x,\ y)\!</math></p> | + | | height="36px" | <p><math>(x,~y)\!</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>((x,\ y))\!</math></p> | + | | height="36px" | <p><math>((x,~y))\!</math></p> |
| |} | | |} |
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| {| align="center" | | {| align="center" |
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− | | height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{not~equal~to}~ y</math></p> |
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− | | height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{equal~to}~ y</math></p> |
| |} | | |} |
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− | | height="36px" | <p><math>\operatorname{not}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ y</math></p> |
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| | height="36px" | <p><math>y\!</math></p> | | | height="36px" | <p><math>y\!</math></p> |
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− | | height="36px" | <p><math>(x\ y)\!</math></p> | + | | height="36px" | <p><math>(x~y)\!</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>(x\ (y))\!</math></p> | + | | height="36px" | <p><math>(x~(y))\!</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>((x)\ y)\!</math></p> | + | | height="36px" | <p><math>((x)~y)\!</math></p> |
| |- | | |- |
| | height="36px" | <p><math>((x)(y))\!</math></p> | | | height="36px" | <p><math>((x)(y))\!</math></p> |
Line 1,059: |
Line 1,099: |
| {| align="center" | | {| align="center" |
| |- | | |- |
− | | height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{not~both}~ x ~\operatorname{and}~ y</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ x ~\operatorname{without}~ y</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p> | + | | height="36px" | <p><math>\operatorname{not}~ y ~\operatorname{without}~ x</math></p> |
| |- | | |- |
− | | height="36px" | <p><math>x\ \operatorname{or}\ y</math></p> | + | | height="36px" | <p><math>x ~\operatorname{or}~ y</math></p> |
| |} | | |} |
| | | | | |
Line 2,360: |
Line 2,400: |
| | | | | |
| <math>\begin{smallmatrix} | | <math>\begin{smallmatrix} |
− | (y) & \operatorname{d}x\ (\operatorname{d}y) & + & | + | (y) & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | (x) & (\operatorname{d}x)\ \operatorname{d}y & + & | + | (x) & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\ | + | ((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\ |
− | y & \operatorname{d}x\ (\operatorname{d}y) & + & | + | y & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | (x) & (\operatorname{d}x)\ \operatorname{d}y & + & | + | (x) & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | (x, y) & \operatorname{d}x\ \operatorname{d}y \\ | + | (x, y) & \operatorname{d}x ~ \operatorname{d}y \\ |
− | (y) & \operatorname{d}x\ (\operatorname{d}y) & + & | + | (y) & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | x & (\operatorname{d}x)\ \operatorname{d}y & + & | + | x & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | (x, y) & \operatorname{d}x\ \operatorname{d}y \\ | + | (x, y) & \operatorname{d}x ~ \operatorname{d}y \\ |
− | y & \operatorname{d}x\ (\operatorname{d}y) & + & | + | y & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | x & (\operatorname{d}x)\ \operatorname{d}y & + & | + | x & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\ | + | ((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\ |
| \end{smallmatrix}</math> | | \end{smallmatrix}</math> |
| |} | | |} |
Line 2,405: |
Line 2,445: |
| | | | | |
| <math>\begin{smallmatrix} | | <math>\begin{smallmatrix} |
− | \operatorname{d}x\ (\operatorname{d}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) & + & | + | \operatorname{d}x~(\operatorname{d}y) & + & |
− | \operatorname{d}x\ \operatorname{d}y \\ | + | \operatorname{d}x~ \operatorname{d}y \\ |
| \end{smallmatrix}</math> | | \end{smallmatrix}</math> |
| |} | | |} |
Line 2,440: |
Line 2,480: |
| | | | | |
| <math>\begin{smallmatrix} | | <math>\begin{smallmatrix} |
− | \operatorname{d}x\ (\operatorname{d}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) & + & | + | \operatorname{d}x ~(\operatorname{d}y) & + & |
− | (\operatorname{d}x)\ \operatorname{d}y \\ | + | (\operatorname{d}x)~ \operatorname{d}y \\ |
| \end{smallmatrix}</math> | | \end{smallmatrix}</math> |
| |} | | |} |
Line 2,475: |
Line 2,515: |
| | | | | |
| <math>\begin{smallmatrix} | | <math>\begin{smallmatrix} |
− | (\operatorname{d}x)\ \operatorname{d}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 & + & | + | (\operatorname{d}x)~\operatorname{d}y & + & |
− | \operatorname{d}x\ \operatorname{d}y \\ | + | \operatorname{d}x ~\operatorname{d}y \\ |
| \end{smallmatrix}</math> | | \end{smallmatrix}</math> |
| |} | | |} |
Line 2,514: |
Line 2,554: |
| | | | | |
| <math>\begin{smallmatrix} | | <math>\begin{smallmatrix} |
− | y & \operatorname{d}x\ (\operatorname{d}y) & + & | + | y & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | x & (\operatorname{d}x)\ \operatorname{d}y & + & | + | x & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\ | + | ((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\ |
− | (y) & \operatorname{d}x\ (\operatorname{d}y) & + & | + | (y) & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | x & (\operatorname{d}x)\ \operatorname{d}y & + & | + | x & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | (x, y) & \operatorname{d}x\ \operatorname{d}y \\ | + | (x, y) & \operatorname{d}x ~ \operatorname{d}y \\ |
− | y & \operatorname{d}x\ (\operatorname{d}y) & + & | + | y & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | (x) & (\operatorname{d}x)\ \operatorname{d}y & + & | + | (x) & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | (x, y) & \operatorname{d}x\ \operatorname{d}y \\ | + | (x, y) & \operatorname{d}x ~ \operatorname{d}y \\ |
− | (y) & \operatorname{d}x\ (\operatorname{d}y) & + & | + | (y) & \operatorname{d}x ~(\operatorname{d}y) & + & |
− | (x) & (\operatorname{d}x)\ \operatorname{d}y & + & | + | (x) & (\operatorname{d}x)~ \operatorname{d}y & + & |
− | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\ | + | ((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\ |
| \end{smallmatrix}</math> | | \end{smallmatrix}</math> |
| |} | | |} |
Line 2,775: |
Line 2,815: |
| | | | | |
| <p><math>\operatorname{d}f = </math></p> | | <p><math>\operatorname{d}f = </math></p> |
− | <p><math>\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y</math></p> | + | <p><math>\partial_x f \cdot \operatorname{d}x ~+~ \partial_y f \cdot \operatorname{d}y</math></p> |
| | | | | |
| <p><math>\operatorname{d}^2 f = </math></p> | | <p><math>\operatorname{d}^2 f = </math></p> |
− | <p><math>\partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y</math></p> | + | <p><math>\partial_{xy} f \cdot \operatorname{d}x \; \operatorname{d}y</math></p> |
− | | <math>\operatorname{d}f|_{x\, y}\!</math> | + | | <math>\operatorname{d}f|_{ x \; y}\!</math> |
− | | <math>\operatorname{d}f|_{x\, (y)}\!</math> | + | | <math>\operatorname{d}f|_{ x \;(y)}\!</math> |
− | | <math>\operatorname{d}f|_{(x)\, y}\!</math> | + | | <math>\operatorname{d}f|_{(x)\; y}\!</math> |
| | <math>\operatorname{d}f|_{(x)(y)}\!</math> | | | <math>\operatorname{d}f|_{(x)(y)}\!</math> |
| |- | | |- |