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<pre>

The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::

EX = !P! x !Q! x d!P! x d!Q! that are indicated

by the difference map Df : EX -> B, namely, the

following six points or singular propositions:

  1.   p  q  dp dq

  2.   p  q  dp (dq)

  3.   p  q (dp) dq

  4.   p (q)(dp) dq

1. &  p & q  &  \operatorname{d}p  & \operatorname{d}q

  5. (p) q  dp (dq)

  6. (p)(q) dp dq

2. &  p & q  &  \operatorname{d}p  & (\operatorname{d}q)

3. &  p & q & (\operatorname{d}p) &  \operatorname{d}q

4. &  p & (q) & (\operatorname{d}p) &  \operatorname{d}q

5. & (p) &  q  &  \operatorname{d}p  & (\operatorname{d}q)

6. & (p) & (q) & \operatorname{d}p  &  \operatorname{d}q

The information borne by <math>\operatorname{D}f</math> should be clear enough from a survey of these six points &mdash; they tell you what you have do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.

as telling you what you have to do from each point

of X in order to change the value borne by f<p, q>

at the point in question, that is, in order to get

to a point where the value of f<p, q> is different

from what it is where you started.

==Note 5==

==Note 5==