Changes

When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.</math>

When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.</math>

For example, take the connection <math>F : \mathbb{B}^2 \to \mathbb{B}</math> such that:

For example, take the connection <math>F : \mathbb{B}^2 \to \mathbb{B}</math> defined as follows.

: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(}~x~,~y~\texttt{)}</math>

| <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(}~x~,~y~\texttt{)}</math>

The connection in question is a boolean function on the variables <math>x, y</math> that returns a value of <math>1</math> just when just one of the pair <math>x, y</math> is not equal to <math>1,</math> or what amounts to the same thing, just when just one of the pair <math>x, y</math> is equal to <math>1.</math>  There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\mathbb{B} = \{ 0, 1 \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \mathrm{GF}(2)</math> that is otherwise known as <math>x + y.</math>

The connection in question is a boolean function on the variables <math>x, y</math> that returns a value of <math>1</math> just when just one of the pair <math>x, y</math> is not equal to <math>1,</math> or what amounts to the same thing, just when just one of the pair <math>x, y</math> is equal to <math>1.</math>  There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\mathbb{B} = \{ 0, 1 \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \mathrm{GF}(2)</math> that is otherwise known as <math>x + y.</math>

The same connection <math>F : \mathbb{B}^2 \to \mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \mathbb{B}^2.</math>  If <math>s</math> is a sentence that denotes the proposition <math>F,</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>\text{“} \, x ~\mathrm{is~not~equal~to}~ y \, \text{”}.</math>  In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:

The same connection <math>F : \mathbb{B}^2 \to \mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \mathbb{B}^2.</math>  If <math>s</math> is a sentence that denotes the proposition <math>F,</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>\text{“} \, x ~\mathrm{is~not~equal~to}~ y \, \text{”}.</math>  In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set.

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

{| align="center" cellpadding="8"

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

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 = \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 = \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 \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 <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 \mathbb{B},</math> and suppose that <math>p, q</math> are indicator functions of the sets <math>P, Q \subseteq X,</math> respectively.&nbsp; In other words, we have the following data.

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

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<math>\begin{matrix}

<math>\begin{matrix}

|}

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Then one has an operator <math>F^\$,</math> the stretch of the connection <math>F</math> over <math>X,</math> and a proposition <math>F^\$ (p, q),</math> the stretch of <math>F</math> to <math>(p, q)</math> on <math>X,</math> with the following properties:

Then one has an operator <math>F^\$,</math> the stretch of the connection <math>F</math> over <math>X,</math> and a proposition <math>F^\$ (p, q),</math> the stretch of <math>F</math> to <math>(p, q)</math> on <math>X,</math> with the following properties.

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

{| align="center" cellpadding="8"

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<math>\begin{array}{ccccl}

<math>\begin{array}{ccccl}

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|}

As a result, the application of the proposition <math>F^\$ (p, q)</math> to each <math>x \in X</math> returns a logical value in <math>\mathbb{B},</math> all in accord with the following equations:

As a result, the application of the proposition <math>F^\$ (p, q)</math> to each <math>x \in X</math> returns a logical value in <math>\mathbb{B},</math> all in accord with the following equations.

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

{| align="center" cellpadding="8"

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<math>\begin{matrix}

<math>\begin{matrix}

|}

|}

For each choice of propositions <math>p</math> and <math>q</math> about things in <math>X,</math> the stretch of <math>F</math> to <math>p</math> and <math>q</math> on <math>X</math> is just another proposition about things in <math>X,</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p</math> and <math>q.</math>  Like any other proposition about things in <math>X,</math> it indicates a subset of <math>X,</math> namely, the fiber that is variously described in the following ways:

For each choice of propositions <math>p</math> and <math>q</math> about things in <math>X,</math> the stretch of <math>F</math> to <math>p</math> and <math>q</math> on <math>X</math> is just another proposition about things in <math>X,</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p</math> and <math>q.</math>  Like any other proposition about things in <math>X,</math> it indicates a subset of <math>X,</math> namely, the fiber that is variously described in the following ways.

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

{| align="center" cellpadding="8"

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<math>\begin{array}{lll}

<math>\begin{array}{lll}