Line 662:
Line 662:
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(p, q) = pq.\!</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(p, q) = pq.\!</math>
−
<pre>
+
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
−
To begin we need to formulate a suitably generic
−
definition of the extended universe of discourse:
−
Relative to an initial domain X = X_1 x ... x X_k,
+
{| align="center" cellpadding="6" width="90%"
−
+
|
−
EX = X x dX = X_1 x ... x X_k x dX_1 x ... x dX_k.
+
<math>\begin{array}{cccl}
+
\text{Let} &
+
X
+
& = &
+
X_1 \times \ldots \times X_k.
+
\\[6pt]
+
\text{Let} &
+
\operatorname{d}X
+
& = &
+
\operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.
+
\\[6pt]
+
\text{Then} &
+
\operatorname{E}X
+
& = &
+
X \times \operatorname{d}X
+
\\[6pt]
+
&
+
& = & X_1 \times \ldots \times X_k ~\times~ \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.
+
\end{array}</math>
+
|}
+
<pre>
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