Changes

: <math>\operatorname{d}A\ =\ \prod_{i=1}^n \operatorname{d}A_i\ =\ \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>

: <math>\operatorname{d}A\ =\ \prod_{i=1}^n \operatorname{d}A_i\ =\ \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>

Here, <math>\operatorname{d}A_i</math> is a set of two differential propositions, <math>\operatorname{d}A_i = \{(\operatorname{d}a_i), \operatorname{d}a_i\},</math> where <math>(\operatorname{d}a_i)</math> is a proposition with the logical value of "<math>\mbox{not}\ \operatorname{d}a_i</math>".  Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{(\operatorname{d}a_i), \operatorname{d}a_i\} \cong \{0, 1\}.</math>  However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},</math> whose intension may be indicated as follows:

Here, <math>\operatorname{d}A_i</math> is a set of two differential propositions, <math>\operatorname{d}A_i = \{(\operatorname{d}a_i), \operatorname{d}a_i\},</math> where <math>(\operatorname{d}a_i)</math> is a proposition with the logical value of "<math>\mbox{not}\ \operatorname{d}a_i</math>".  Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> operating under the ordered correspondence <math>\{(\operatorname{d}a_i), \operatorname{d}a_i\} \cong \{0, 1\}.</math>  However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},</math> whose intension may be indicated as follows:

: '''D''' = {(d''a''_''i''),&nbsp;d''a''_''i''} = {same,&nbsp;different} = {stay,&nbsp;change} = {stop,&nbsp;step}.

: <math>\mathbb{D} = \{(\operatorname{d}a_i), \operatorname{d}a_i\} = \{\mbox{same}, \mbox{different}\} = \{\mbox{stay}, \mbox{change}\} = \{\mbox{stop}, \mboxx{step}\}.</math>

Viewed within a coordinate representation, spaces of type '''B'''^''n'' and '''D'''^''n'' may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

Viewed within a coordinate representation, spaces of type '''B'''^''n'' and '''D'''^''n'' may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.