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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 18:25, 4 June 2008
, 18:25, 4 June 2008→Formal development: sub[a/the] + spacing
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:
:* 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 the ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:
:* 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><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></p>
::: <p><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></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 the ''first order extended basis'', <math>\operatorname{E}\mathcal{A},</math> defined as follows:
:* 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><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></p>
::: <p><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></p>