Changes

By inspection, it is fairly easy to understand <math>\operatorname{D}f</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by <math>f(x, y).\!</math>

By inspection, it is fairly easy to understand <math>\operatorname{D}f</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by <math>f(x, y).\!</math>

We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&rarr;&nbsp;'''B''' that is commonly known as the conjunction ''xy''.  We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp;&times;&nbsp;''dY''&nbsp;&rarr;&nbsp;'''B'''. .Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp;&times;&nbsp;''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' = .''X''&nbsp;&times;&nbsp;''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX''&nbsp;&times;&nbsp;''dY''.

We have been studying the action of the difference operator <math>\operatorname{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>xy.\!</math> We described <math>\operatorname{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\operatorname{D}f : X \times Y \times \operatorname{d}X \times \operatorname{d}Y \to \mathbb{B}.</math>  Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of <math>\operatorname{D}f</math> distribute within the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> we can depict <math>\operatorname{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U = X \times Y</math> and whose arrows are labeled with the elements of <math>\operatorname{d}U = \operatorname{d}X \times \operatorname{d}Y.</math>

<pre>

<pre>