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Written as a string, this is just the concatenation "<math>x\ y\!</math>".

Written as a string, this is just the concatenation "<math>xy\!</math>".

The proposition <math>x y\!</math> may be taken as a boolean function <math>f(x, y)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that 0 means <math>false\!</math> and 1 means <math>true.\!</math>

The proposition <math>xy\!</math> may be taken as a boolean function <math>f(x, y)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that 0 means <math>false</math> and 1 means <math>true.</math>

In this style of graphical representation, the value <math>true\!</math> looks like a blank label and the value <math>false\!</math> looks like an edge.

In this style of graphical representation, the value <math>true</math> looks like a blank label and the value <math>false</math> looks like an edge.

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Back to the proposition <math>x y.\!</math>  Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>x y\!</math> is true, as pictured:

Back to the proposition <math>xy.\!</math>  Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>xy\!</math> is true, as pictured:

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Now ask yourself: .What is the value of the proposition ''xy'' at a distance of ''dx'' and ''dy'' from the cell ''xy'' where you are standing?

Now ask yourself: What is the value of the proposition <math>xy\!</math> at a distance of <math>\operatorname{d}x</math> and <math>\operatorname{d}y</math> from the cell <math>xy\!</math> where you are standing?

Don't think about it -- just compute:

Don't think about it &mdash; just compute:

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However you draw it, these expressions follow because the expression ''x'' + ''dx'', where the plus sign indicates (mod 2) addition in '''B''', and thus corresponds to an exclusive-or in logic, parses to a graph of the following form:

However you draw it, these expressions follow because the expression <math>x + \operatorname{d}x,</math> where the plus sign indicates (mod 2) addition in <math>\mathbb{B},</math> and thus corresponds to an exclusive-or in logic, parses to a graph of the following form:

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