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MyWikiBiz, Author Your Legacy — Monday March 17, 2025
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<pre>
−Date: 30 Nov 2008, 2:00 AM
−Author: Jon Awbrey
−Subject: Re: logical equivalence problem
−o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o+required to show: ~(p <=> q) is equivalent to (~q) <=> p+
−in logical graphs, the required equivalence looks like this:
+we have a theorem that says:+see: http://www.mywikibiz.com/Logical_graph#C2.__Generation_theorem+applying this twice to the left hand side of the required equation:+by collection, the reverse of distribution, we get:+but this is the same result that we get from one application of+ http://stderr.org/pipermail/inquiry/
− since I know it preserves the trees.
−
−o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−</pre>
* [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].
−Required to show: ~(p <=> q) is equivalent to (~q) <=> p.
−In logical graphs, the required equivalence looks like this:
−<pre>
q o o p q o
q o o p q o
| | |
| | |
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| \ /
| \ /
@ = @
@ = @
+</pre>
−We have a theorem that says:
+<pre>
y o xy o
y o xy o
| |
| |
x @ = x @
x @ = x @
+</pre>
−See [http://www.mywikibiz.com/Logical_graph#C2.__Generation_theorem Logical Graph : C<sub>2</sub>. Generation Theorem].
−Applying this twice to the left hand side of the required equation, we get:
+<pre>
q o o p pq o o pq
q o o p pq o o pq
| | | |
| | | |
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| |
| |
@ = @
@ = @
+</pre>
−By collection, the reverse of distribution, we get:
+<pre>
p q
p q
o o
o o
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\ /
\ /
@
@
+</pre>
−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
−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.
−===Discussion===
===Discussion===
