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Math Help - Isomorphic Structures

  1. #1
    drg
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    Isomorphic Structures

    Hi everyone,

    This problem is about first order logic.

    Given a structure A and a bijection g with domain |A| (i.e. the universe of A), I have to show that there is a unique structure B such that g is an isomorphism of A onto B.

    Since A is isomorphic to itself, this basically comes down to showing that every structure is unique up to isomorphism. While I would feel somewhat comfortable showing this for certain types of sets, I'm having trouble seeing how to go about doing this with structure.

    Any help would be appreciated.
    Last edited by drg; November 7th 2012 at 03:27 PM.
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  2. #2
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    Re: Isomorphic Structures

    Quote Originally Posted by drg View Post
    HGiven a structure A and a bijection g with domain |A|, I have to show that there is a unique structure B such that g is an isomorphism of A onto B.
    .
    ..a structure A what does that term mean?
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  3. #3
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    Re: Isomorphic Structures

    Quote Originally Posted by drg View Post
    Given a structure A and a bijection g with domain |A| (i.e. the universe of A), I have to show that there is a unique structure B such that g is an isomorphism of A onto B.
    I believe this is a better link that explains what a structure is in the context of mathematical logic.

    If I understand correctly, g is an isomorphism between A and B if

    (1) g is a bijection between |A| and |B|,
    (2) for every functional symbol f and every \vec{a}\in |A| it is the case that g(f^A(\vec{a}))=f^B(g(\vec{a})), and
    (3) for every predicate symbol R and every \vec{a}\in |A| it is the case that R^A(\vec{a}) iff R^B(g(\vec{a})).

    The universe of B must be the image of g. We need to show that there exists one and only one interpretation of functional and predicate symbols on |B| such that g is an isomorphism. Well, this interpretation is uniquely determined by properties (2) and (3).
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    drg
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    Re: Isomorphic Structures

    Quote Originally Posted by emakarov View Post
    I believe this is a better link that explains what a structure is in the context of mathematical logic.

    If I understand correctly, g is an isomorphism between A and B if

    (1) g is a bijection between |A| and |B|,
    (2) for every functional symbol f and every \vec{a}\in |A| it is the case that g(f^A(\vec{a}))=f^B(g(\vec{a})), and
    (3) for every predicate symbol R and every \vec{a}\in |A| it is the case that R^A(\vec{a}) iff R^B(g(\vec{a})).

    The universe of B must be the image of g. We need to show that there exists one and only one interpretation of functional and predicate symbols on |B| such that g is an isomorphism. Well, this interpretation is uniquely determined by properties (2) and (3).
    Yea my bad that is the article I had in mind.

    Thanks that actually helped, even though I was sort of missing the point and realized that the problem basically says: given bijection g there IS a UNIQUE structure B isomorphic to A UNDER g. Might sound redundant but it makes sense to me now.
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