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Back to the proposition <math>xy.\!</math>  Imagine yourself standing

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 shown here:

in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>xy\!</math> is true, as shown here:

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

==Note 4==

==Note 4==

We have been studying the action of the difference operator <math>\operatorname{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>x \cdot y.</math>  We described <math>\operatorname{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\operatorname{D}f : X \times Y \times \operatorname{d}X \times \operatorname{d}Y \to \mathbb{B}.</math>  Abstracting from the augmented venn diagram that illustrates how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute within the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> we can depict <math>\operatorname{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U =  X \times Y</math> and whose arrows are labeled with the elements of <math>\operatorname{d}U = \operatorname{d}X \times \operatorname{d}Y.</math>

We have been studying the action of the difference operator <math>\operatorname{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>x \cdot y.</math>  We described <math>\operatorname{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\operatorname{D}f : X \times Y \times \operatorname{d}X \times \operatorname{d}Y \to \mathbb{B}.</math>  Abstracting from the augmented venn diagram that illustrates how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute within the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> we can depict <math>\operatorname{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U =  X \times Y</math> and whose arrows are labeled with the elements of <math>\operatorname{d}U = \operatorname{d}X \times \operatorname{d}Y.\!</math>

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

==Note 6==

==Note 6==

To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.

To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math>  A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.

Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.

Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.

(x)~y~

(x)~y~

\\[4pt]

\\[4pt]

(x)~~~

(x)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])

\\[4pt]

\\[4pt]

~x~(y)

~x~(y)

\\[4pt]

\\[4pt]

~~~(y)

[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(y)

\\[4pt]

\\[4pt]

(x,~y)

(x,~y)

((x,~y))

((x,~y))

\\[4pt]

\\[4pt]

~~~~~y~~

21:56, 7 December 2014 (UTC)y~~

\\[4pt]

\\[4pt]

~(x~(y))

~(x~(y))

\\[4pt]

\\[4pt]

~~x~~~~~

~~x21:56, 7 December 2014 (UTC)

\\[4pt]

\\[4pt]

((x)~y)~

((x)~y)~

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So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{~~~~}\, {}^{\prime\prime}</math> that is represented by the following matrix:

So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:56, 7 December 2014 (UTC)}\, {}^{\prime\prime}</math> that is represented by the following matrix:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

Back to our current subinstance, the example in support of our first example.  I will try to reconstruct it in a less confusing way.

Back to our current subinstance, the example in support of our first example.  I will try to reconstruct it in a less confusing way.

Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{~~~~}\, {}^{\prime\prime},</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:

Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:56, 7 December 2014 (UTC)}\, {}^{\prime\prime},</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

|}

|}

Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{~~~~}\, {}^{\prime\prime}</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.

Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:56, 7 December 2014 (UTC)}\, {}^{\prime\prime}</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.

Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math>  This is the mode of reading that we call ''multiplying on the left''.

Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math>  This is the mode of reading that we call ''multiplying on the left''.

===Ontology List (Apr&ndash;Jul 2002)===

===Ontology List (Apr&ndash;Jul 2002)===

* http://suo.ieee.org/ontology/thrd28.html#04040

* http://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04040

# http://suo.ieee.org/ontology/msg04040.html

# http://web.archive.org/web/20140406040004/http://suo.ieee.org/ontology/msg04040.html

# http://suo.ieee.org/ontology/msg04041.html

# http://web.archive.org/web/20110612001949/http://suo.ieee.org/ontology/msg04041.html

# http://suo.ieee.org/ontology/msg04045.html

# http://web.archive.org/web/20110612010502/http://suo.ieee.org/ontology/msg04045.html

# http://suo.ieee.org/ontology/msg04046.html

# http://web.archive.org/web/20110612005212/http://suo.ieee.org/ontology/msg04046.html

# http://suo.ieee.org/ontology/msg04047.html

# http://web.archive.org/web/20110612001954/http://suo.ieee.org/ontology/msg04047.html

# http://suo.ieee.org/ontology/msg04048.html

# http://web.archive.org/web/20110612010620/http://suo.ieee.org/ontology/msg04048.html

# http://suo.ieee.org/ontology/msg04052.html

# http://web.archive.org/web/20110612010550/http://suo.ieee.org/ontology/msg04052.html

# http://suo.ieee.org/ontology/msg04054.html

# http://web.archive.org/web/20110612010724/http://suo.ieee.org/ontology/msg04054.html

# http://suo.ieee.org/ontology/msg04055.html

# http://web.archive.org/web/20110612000847/http://suo.ieee.org/ontology/msg04055.html

# http://suo.ieee.org/ontology/msg04067.html

# http://web.archive.org/web/20110612001959/http://suo.ieee.org/ontology/msg04067.html

# http://suo.ieee.org/ontology/msg04068.html

# http://web.archive.org/web/20110612010507/http://suo.ieee.org/ontology/msg04068.html

# http://suo.ieee.org/ontology/msg04069.html

# http://web.archive.org/web/20110612002014/http://suo.ieee.org/ontology/msg04069.html

# http://suo.ieee.org/ontology/msg04070.html

# http://web.archive.org/web/20110612010701/http://suo.ieee.org/ontology/msg04070.html

# http://suo.ieee.org/ontology/msg04072.html

# http://web.archive.org/web/20110612003540/http://suo.ieee.org/ontology/msg04072.html

# http://suo.ieee.org/ontology/msg04073.html

# http://web.archive.org/web/20110612005229/http://suo.ieee.org/ontology/msg04073.html

# http://suo.ieee.org/ontology/msg04074.html

# http://web.archive.org/web/20110610153117/http://suo.ieee.org/ontology/msg04074.html

# http://suo.ieee.org/ontology/msg04077.html

# http://web.archive.org/web/20110612010555/http://suo.ieee.org/ontology/msg04077.html

# http://suo.ieee.org/ontology/msg04079.html

# http://web.archive.org/web/20110612001918/http://suo.ieee.org/ontology/msg04079.html

# http://suo.ieee.org/ontology/msg04080.html

# http://web.archive.org/web/20110612005244/http://suo.ieee.org/ontology/msg04080.html

# http://suo.ieee.org/ontology/msg04268.html

# http://web.archive.org/web/20110612005249/http://suo.ieee.org/ontology/msg04268.html

# http://suo.ieee.org/ontology/msg04269.html

# http://web.archive.org/web/20110612010626/http://suo.ieee.org/ontology/msg04269.html

# http://suo.ieee.org/ontology/msg04272.html

# http://web.archive.org/web/20110612000853/http://suo.ieee.org/ontology/msg04272.html

# http://suo.ieee.org/ontology/msg04273.html

# http://web.archive.org/web/20110612010514/http://suo.ieee.org/ontology/msg04273.html

# http://suo.ieee.org/ontology/msg04290.html

# http://web.archive.org/web/20110612002235/http://suo.ieee.org/ontology/msg04290.html

===Inquiry List (May & Jul 2004)===

===Inquiry List (May&ndash;Jul 2004)===

* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400

* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400

* http://stderr.org/pipermail/inquiry/2004-July/thread.html#1685

* http://stderr.org/pipermail/inquiry/2004-July/thread.html#1685

===NKS Forum (May & Jul 2004)===

===NKS Forum (May&ndash;Jul 2004)===

* http://forum.wolframscience.com/archive/topic/420-1.html

* http://forum.wolframscience.com/archive/topic/420.html

* http://forum.wolframscience.com/printthread.php?threadid=420

* http://forum.wolframscience.com/printthread.php?threadid=420

* http://forum.wolframscience.com/showthread.php?threadid=420

* http://forum.wolframscience.com/showthread.php?threadid=420