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Directory:Jon Awbrey/Papers/Cactus Language (view source)

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→‎Stretching Exercises: mathematical markup + copy editing

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

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

<pre>

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~~This connection is the boolean function on a couple of variables x, y~~

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~~that yields a value of %1% if and only if just one of x, y is not %1%,~~

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~~that is, if and only if just one of x, y is %1%. There is clearly an~~

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~~isomorphism between this connection, viewed as an operation on the~~

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~~boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary~~

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~~values x, y in !B! = GF(2) that is otherwise known as "x + y".~~

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The same connection F : %B%^2 -> %B% can also be read as a proposition

about things in the universe X = %B%^2. If S is a sentence that denotes

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