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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 19:22, 18 August 2009
, 19:22, 18 August 2009→Analysis of contingent propositions: convert graphic
It remains to show that the right hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}</math> is logically equivalent to the DNF just obtained. The needed chain of equations is as follows:
It remains to show that the right hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}</math> is logically equivalent to the DNF just obtained. The needed chain of equations is as follows:
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{| align="center" cellpadding="8"
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-0.jpg|500px]]
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-1.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
| | |
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-4.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cast Q.jpg|500px]]
|-
| \ / |
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
| \ / |
|-
| \ / |
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| @ |
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| q r |
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-7.jpg|500px]]
| | \ |
|-
| [[Image:Equational Inference Bar -- Cast R.jpg|500px]]
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-8.jpg|500px]]
| \ / |
|-
| \ / |
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-9.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
|}
| q r |
| o |
| \ / |
| \ / |
| \ / |
| p o-------o---o p |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| p o-------o---o p |
| \ / |
| \ / |
| o |
| r o-------o---o r |
| \ / |
| \ / o |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| r o-------o---o r |
| \ / |
| \ / o |
| \ / |
| \ / |
| \ / |
| (33)
| (33)
|}
|}