Results 1 to 4 of 4

Math Help - Determine whether alpha |- gamma

  1. #1
    Junior Member
    Joined
    Jul 2011
    Posts
    26

    Determine whether alpha |- gamma

    Hi,

    Please could someone help me with the following question?

    Let \alpha : \exists x( \neg A(x) \vee B(x))
    Let \gamma : \neg \exists x A(x) \vee \exists xB(x)
    Determine whether \alpha \vdash \gamma.

    I've tried both writing a formal proof to see if it is provable and creating a model to see if it's not provable, but I don't seem to be able to reach a conclusion.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jul 2011
    Posts
    26

    Re: Determine whether alpha |- gamma

    Quote Originally Posted by dwally89 View Post
    Hi,

    Please could someone help me with the following question?

    Let \alpha : \exists x( \neg A(x) \vee B(x))
    Let \gamma : \neg \exists x A(x) \vee \exists xB(x)
    Determine whether \alpha \vdash \gamma.

    I've tried both writing a formal proof to see if it is provable and creating a model to see if it's not provable, but I don't seem to be able to reach a conclusion.

    Thanks
    Would the following model show that \alpha does not prove \gamma?

    Let M=\{ 1,2\}
    Let M \supset A = \{ 1 \}
    Let M \supset B = \{ \}

    \alpha can be satisfied by letting x=2, as the first first clause \neg A(x) will be true.
    \gamma on the other hand can never be satisfied.
    The first clause \neg \exists xA(x) can be made false by setting x=1, and the second clause B(x) is always false.

    Is this ok?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785

    Re: Determine whether alpha |- gamma

    Yes. The formula \exists x( \neg A(x) \vee B(x)) is equivalent to (\exists x\,\neg A(x)) \vee (\exists x\,B(x)), while (\neg \exists x\,A(x)) \vee (\exists x\,B(x)) is equivalent to (\forall x\,\neg A(x))\lor(\exists x\,B(x)). So \alpha does not derive \gamma because \exists x\,\neg A(x) does not derive \forall x\,\neg A(x). A countermodel should have \neg A(x) true for some x but not for all x, which is what you have done.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jul 2011
    Posts
    26

    Re: Determine whether alpha |- gamma

    Quote Originally Posted by emakarov View Post
    Yes. The formula \exists x( \neg A(x) \vee B(x)) is equivalent to (\exists x\,\neg A(x)) \vee (\exists x\,B(x)), while (\neg \exists x\,A(x)) \vee (\exists x\,B(x)) is equivalent to (\forall x\,\neg A(x))\lor(\exists x\,B(x)). So \alpha does not derive \gamma because \exists x\,\neg A(x) does not derive \forall x\,\neg A(x). A countermodel should have \neg A(x) true for some x but not for all x, which is what you have done.
    Thanks :-)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Let alpha be a real number, alpha > -1 now show that
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: November 27th 2011, 08:14 PM
  2. Replies: 1
    Last Post: November 14th 2011, 03:48 AM
  3. Gamma - Gamma parameter estimation EM algorithm
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: July 24th 2010, 10:53 AM
  4. find g(X) that transforms folded normal into gamma(alpha,beta)
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: July 22nd 2010, 12:07 PM
  5. Replies: 0
    Last Post: March 30th 2008, 01:44 PM

Search Tags


/mathhelpforum @mathhelpforum