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| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]

| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]

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| <math>\text{Figure 25-1.  Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Figure 25-1.  Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other.  The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true.  The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math>  If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure&nbsp;25-2.

In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other.  The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true.  The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math>  If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure&nbsp;25-2.

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| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]

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<pre>

<math>\begin{array}{rcccccc}

& = &  p  & \cdot & q & \cdot & ((\operatorname{d}p)(\operatorname{d}q))

\\[4pt]

& + &  p  & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~

|          /                    \ /                    \          |

\\[4pt]

|          / P                    o                    Q \         |

& + & (p) & \cdot &  q  & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~

|        /                      / \                       \         |

\\[4pt]

|        /                      /  \                       \       |

& + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~

|      /                      /    \                       \       |

\end{array}</math>

|      /                      /      \                       \     |

|    /                      /        \                       \     |

|    |            (dp) dq  |          |  dp (dq)            |    |

|    \                       \   |    /                      /    |

|      \                       \   |  /                      /      |

|      \                       \ |  /                      /      |

|        \                       \ | /                      /        |

|        \                       \|/                      /        |

|          \                       |                      /          |

|          \                     /|\                    /          |

</pre>

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==Note 26==

==Note 26==