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==Stretching Exercises==

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~~Taking up the preceding~~ arrays of ~~particular~~ connections, namely, the boolean functions ~~on two or less variables~~, ~~it possible~~ to ~~illustrate~~ the use of the stretch operation in a variety of concrete cases.

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The arrays of boolean connections described above, namely, the boolean functions <math>F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>k\!</math> in <math>\{ 0, 1, 2 \},\!</math> supply enough material to demonstrate the use of the stretch operation in a variety of concrete cases.

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For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table 18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>

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For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table 18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math>, or else the indicators of sets contained in <math>X.\!</math>

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Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.

<pre>

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~~Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2,~~

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~~and the stretch of the connection F to #f# on X amounts to a proposition~~

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~~F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as~~

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~~the "stretch of F to p and q". If one is concerned with many different~~

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~~propositions about things in X, or if one is abstractly indifferent to~~

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~~the particular choices for p and q, then one can detach the operator~~

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~~F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X",~~

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~~and consider it in isolation from any concrete application.~~

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When the "cactus notation" is used to represent boolean functions,

a single "$" sign at the end of the expression is enough to remind

Jon Awbrey

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