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Math Help - Question on duality theorem[Mathematical logic]

  1. #1
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    Question on duality theorem[Mathematical logic]

    Hi everybody.
    in VanDallen's "Logic and structure", he defines a "duality mapping" d from PROP to PROP as follows(note that by !phi i mean negation of phi):

    d(phi)=phi for atomic phi.
    d(phi ^ psi)=d(phi) or d(psi)
    d(phi or psi)=d(phi) ^ d(psi)
    d(!phi)= !(d(phi))
    -----------------
    then he proves the following theorem(duality theorem):[note by p == q, i mean p and q are equivalent]
    phi == psi iff d(phi) == d(psi) [in fluent English, if two propositions are equivalence, then so are their duals.]

    -----------------

    Now my question:


    We know that (p ^ (!p or q)) == q. Then why their duals are not so? i.e., why the following does not hold? : (p or (!p ^ q)) == q.


    thanks.
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  2. #2
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    Re: Question on duality theorem[Mathematical logic]

    (p ^ (!p or q)) implies q but is not equivalent to q.
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    Member Mathelogician's Avatar
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    Re: Question on duality theorem[Mathematical logic]

    Hehe you are right... I was sleepy when created the example.
    Thanks.
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