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====Note 5====

====Note 5====

The enlargement operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(x, y) = xy.\!</math>

The ''enlargement'' or ''shift'' operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(x, y) = xy.\!</math>

A suitably generic definition of the extended universe of discourse is afforded by the following set-up:

A suitably generic definition of the extended universe of discourse is afforded by the following set-up:

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For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the (first order) ''enlargement'' of <math>f\!</math> is the

For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the (first order) ''enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}U \to \mathbb{B}</math> that is defined by the following equation:

proposition <math>\operatorname{E}f : \operatorname{E}U \to \mathbb{B}</math> that is defined by the following equation:

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<pre>

The ''differential variables'' <math>\operatorname{d}x_j</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math>  It is conventional to distinguish the (first order) differential variables with the operative prefix "<math>\operatorname{d}</math>", but this is purely optional.  It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.

It should be noted that the so-called "differential variables" dx_j

are really just the same kind of boolean variables as the other x_j.

It is conventional to give the additional variables these brands of

inflected names, but whatever extra connotations we might choose to

attach to these syntactic conveniences are wholly external to their

purely algebraic meanings.

For the example f(x, y) = xy, we obtain:

In the case of logical conjunction, <math>f(x, y) = xy,\!</math> the computation of the enlargement <math>\operatorname{E}f</math> begins as follows:

Ef(x, y, dx, dy)   =   (x + dx)(y + dy).

| <math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) ~=~ (x + \operatorname{d}x)(y + \operatorname{d}y).</math>

Given that this expression uses nothing more than the "boolean ring"

Given that this expression uses nothing more than the "boolean ring"

operations of addition (+) and multiplication (·), it is permissible

operations of addition (+) and multiplication (·), it is permissible