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120 bytes removed ,  22:20, 2 December 2008
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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
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Required to show:  ~(p <=> q) is equivalent to (~q) <=> p.
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required to show:  ~(p <=> q) is equivalent to (~q) <=> p
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In logical graphs, the required equivalence looks like this:
 
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in logical graphs, the required equivalence looks like this:
      +
<pre>
 
       q o  o p          q o
 
       q o  o p          q o
 
         |  |              |
 
         |  |              |
Line 37: Line 32:  
           |                  \ /  
 
           |                  \ /  
 
           @        =        @
 
           @        =        @
 +
</pre>
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we have a theorem that says:
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We have a theorem that says:
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<pre>
 
         y o                xy o
 
         y o                xy o
 
           |                  |
 
           |                  |
 
         x @        =        x @
 
         x @        =        x @
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</pre>
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see: http://www.mywikibiz.com/Logical_graph#C2.__Generation_theorem
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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:
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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
 
         |  |              |  |
 
         |  |              |  |
Line 55: Line 54:  
           |                  |
 
           |                  |
 
           @        =        @
 
           @        =        @
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</pre>
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by collection, the reverse of distribution, we get:
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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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         \ /
 
         \ /
 
           @
 
           @
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</pre>
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but this is the same result that we get from one application of
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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.
      
QED
 
QED
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Jon Awbrey
 
Jon Awbrey
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PS.  I will copy this to the Inquiry List:
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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>
      
===Discussion===
 
===Discussion===
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