Changes

To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:

To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:

{| align="center" cellspacing="20" style="text-align:center; width:90%"

{| align="center" cellspacing="20" style="text-align:center"

| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]

| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]

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|-

Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

{| align="center" cellspacing="10"

{| align="center" cellspacing="10" style="text-align:center"

| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]

| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]

|}

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<math>\begin{array}{rcccccc}

<math>\begin{array}{rcccccc}