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| * [http://mathforum.org/kb/plaintext.jspa?messageID=6514666 Solution posted by Jon Awbrey, working in the medium of logical graphs]. | | * [http://mathforum.org/kb/plaintext.jspa?messageID=6514666 Solution posted by Jon Awbrey, working in the medium of logical graphs]. |
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− | <pre>
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− | Date: 30 Nov 2008, 2:00 AM
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− | Author: Jon Awbrey
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− | Subject: Re: logical equivalence problem
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Required to show: ~(p <=> q) is equivalent to (~q) <=> p. |
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− | required to show: ~(p <=> q) is equivalent to (~q) <=> p
| + | In logical graphs, the required equivalence looks like this: |
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− | in logical graphs, the required equivalence looks like this:
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| + | <pre> |
| q o o p q o | | q o o p q o |
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| | \ / | | | \ / |
| @ = @ | | @ = @ |
| + | </pre> |
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− | we have a theorem that says:
| + | We have a theorem that says: |
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| + | <pre> |
| y o xy o | | y o xy o |
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| x @ = x @ | | x @ = x @ |
| + | </pre> |
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− | see: http://www.mywikibiz.com/Logical_graph#C2.__Generation_theorem
| + | See [http://www.mywikibiz.com/Logical_graph#C2.__Generation_theorem Logical Graph : C<sub>2</sub>. Generation Theorem]. |
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− | applying this twice to the left hand side of the required equation:
| + | Applying this twice to the left hand side of the required equation, we get: |
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| + | <pre> |
| q o o p pq o o pq | | q o o p pq o o pq |
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| @ = @ | | @ = @ |
| + | </pre> |
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− | by collection, the reverse of distribution, we get:
| + | By collection, the reverse of distribution, we get: |
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| + | <pre> |
| p q | | p q |
| o o | | o o |
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| \ / | | \ / |
| @ | | @ |
| + | </pre> |
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− | but this is the same result that we get from one application of
| + | But this is the same result that we get from one application of double negation to the right hand side of the required equation. |
− | double negation to the right hand side of the required equation. | |
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| QED | | QED |
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| Jon Awbrey | | Jon Awbrey |
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− | PS. I will copy this to the Inquiry List: | + | PS. I will copy this to the [http://stderr.org/pipermail/inquiry/ Inquiry List], since I know it preserves the trees. |
− | http://stderr.org/pipermail/inquiry/
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− | since I know it preserves the trees.
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− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− | </pre>
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| ===Discussion=== | | ===Discussion=== |