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Revision as of 04:22, 27 December 2011
, 04:22, 27 December 2011Move [Expository Examples] to Talk Page
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=Work In Progress=
==4. Expository Examples==
'''(Work In Progress)'''
Consider the logical proposition represented by the following venn diagram:
<center><pre>
o-----------------------------------------------------------o
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . .o-------------o. . . . . . . . . . . |
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| . . . . . . . .o. . . . . . . . . . . . .o. . . . . . . . |
| . . . . . . . .|. . . . . . U . . . . . .|. . . . . . . . |
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| . . . . . . . .|. . . . . . . . . . . . .|. . . . . . . . |
| . . . . . . o--o----------o . o----------o--o . . . . . . |
| . . . . . ./. . \%%%%%%%%%%\./%%%%%%%%%%/ . .\. . . . . . |
| . . . . . / . . .\%%%%%%%%%%o%%%%%%%%%%/. . . \ . . . . . |
| . . . . ./. . . . \%%%%%%%%/%\%%%%%%%%/ . . . .\. . . . . |
| . . . . / . . . . .\%%%%%%/%%%\%%%%%%/. . . . . \ . . . . |
| . . . ./. . . . . . \%%%%/%%%%%\%%%%/ . . . . . .\. . . . |
| . . . o . . . . . . .o--o-------o--o. . . . . . . o . . . |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| . . . | . . . .V. . . . |%%%%%%%| . . . .W. . . . | . . . |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| . . . o . . . . . . . . o%%%%%%%o . . . . . . . . o . . . |
| . . . .\. . . . . . . . .\%%%%%/. . . . . . . . ./. . . . |
| . . . . \ . . . . . . . . \%%%/ . . . . . . . . / . . . . |
| . . . . .\. . . . . . . . .\%/. . . . . . . . ./. . . . . |
| . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| . . . . . .\. . . . . . . ./.\. . . . . . . ./. . . . . . |
| . . . . . . o-------------o . o-------------o . . . . . . |
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o-----------------------------------------------------------o
</pre>
'''Figure 1. Proposition''' <math>q : X \to \mathbb{B}</math>
</center>
The following language is useful in describing the facts represented by the venn diagram.
* The universe of discourse is a set, <math>X,\!</math> represented by the area inside the large rectangle.
* The boolean domain is a set of two elements, <math>\mathbb{B} = \{ 0, 1 \},</math> represented by the two distinct shadings of the regions inside the rectangle.
* According to the conventions observed in this context, the algebraic value 0 is interpreted as the logical value <math>\operatorname{false}</math> and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value <math>\operatorname{true}</math> and represented by the darker shading.
* The universe of discourse <math>X\!</math> is the domain of three functions <math>u, v, w : X \to \mathbb{B}</math> called ''basic'', ''coordinate'', or ''simple'' propositions.
* As with any proposition, <math>p : X \to \mathbb{B},</math> a simple proposition partitions <math>X\!</math> into two fibers, the fiber of 0 under <math>p,\!</math> defined as <math>p^{-1}(0) \subseteq X,</math> and the fiber of 1 under <math>p,\!</math> defined as <math>p^{-1}(1) \subseteq X.</math>
* Each coordinate proposition is represented by a "circle", or a simple closed curve, that divides the rectangular region into the region exterior to the circle, representing the fiber of 0 under <math>p,\!</math> and the region interior to the circle, representing the fiber of 1 under <math>p.\!</math>
* The fibers of 1 under the propositions <math>u, v, w\!</math> are the respective subsets <math>U, V, W \subseteq X.</math>
'''…'''
=Material To Be Collated=
=Material To Be Collated=
