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====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 E, also known as the "shift operator",

has many interesting and very useful properties in its own right,

so let us not fail to observe a few of the more salient features

that play out on the surface of our simple example, f(x, y) = xy.

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

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

Let U = X_1 x ... x X_k and EU = U x dU = X_1 x ... x X_k x dX_1 x ... x dX_k.

Let <math>U = X_1 \times \ldots \times X_k</math> and <math>\operatorname{E}U = U \times \operatorname{d}U = X_1 \times \ldots \times X_k \times \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.</math>

For a proposition f : X_1 x ... x X_k -> B,

For a proposition f : X_1 x ... x X_k -> B,

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

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