Changes

This covers the properties of the connection <math>F(x, y) = \underline{(}~x~,~y~\underline{)},</math> treated as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math>  Staying with this same connection, it is time to demonstrate how it can be "stretched" to form an operator on arbitrary propositions.

This covers the properties of the connection <math>F(x, y) = \underline{(}~x~,~y~\underline{)},</math> treated as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math>  Staying with this same connection, it is time to demonstrate how it can be "stretched" to form an operator on arbitrary propositions.

To continue the exercise, let <math>p\!</math> and <math>q\!</math> be arbitrary propositions about things in the universe <math>X,\!</math> that is, maps of the form <math>p, q : X \to \underline\mathbb{B},</math> and suppose that <math>p, q\!</math> are indicator functions of the sets <math>P, Q \subseteq X,</math> respectively.  In other words, we have the following data:

To continue the exercise, let p and q be arbitrary propositions about

things in the universe X, that is, maps of the form p, q : X -> %B%,

and suppose that p, q are indicator functions of the sets P, Q c X,

respectively.  In other words, one has the following set of data:

|   p    =       -{P}-        :  X -> %B%

{| align="center" cellpadding="8" width="90%"

|

|

|  q     =       -{Q}-        :   X -> %B%

| <p, q>  = < -{P}- , -{Q}- >  : (X -> %B%)^2

q

& = &

\upharpoonleft Q \upharpoonright

& : &

X \to \underline\mathbb{B}

(p, q)

& = &

(\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright)

& : &

(X \to \underline\mathbb{B})^2

Then one has an operator F^$, the stretch of the connection F over X,

Then one has an operator F^$, the stretch of the connection F over X,

and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with

and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with