Changes

====Note 5====

====Note 5====

The enlargement operator <math>\operatorname{E},</math> also known as the ''shift operator'', has many interesting and useful properties in its own right, so let's examine a few of the more salient features that play out on the surface of our simple example, <math>f(x, y) = xy.\!</math>

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>

Introduce a suitably generic definition of the extended universe of discourse:

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

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

& = &

& = &

X_1 \times \ldots \times X_k.

X_1 \times \ldots \times X_k.

\\[6pt]

\\[6pt]

\text{Then} & \operatorname{E}U

\text{Then} & \operatorname{E}U

|}

|}

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 f : X_1 x ... x X_k -> B,

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

the (first order) 'enlargement' of f is the

proposition Ef : EU -> B that is defined by:

Ef(x_1, ..., x_k, dx_1, ..., dx_k) =  f(x_1 + dx_1, ..., x_k + dx_k).

| <math>\operatorname{E}f(x_1, \ldots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) ~=~ f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k).</math>

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

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.

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