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Math Help - Set theory question

  1. #1
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    Set theory question

    Could anyone help me with the following question:

    Let R be a relation. Show that domR={x : ∃y ([x, y] ∈ R)}, is a set (where [x,y] is and ordered pair.
    Also, let a and b be sets, Prove that there exists a set whose members are exactly the functions with domain a and codomain b

    Many thanks.
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  2. #2
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by KSM08 View Post
    Could anyone help me with the following question:

    Let R be a relation. Show that domR={x : ∃y ([x, y] ∈ R)}, is a set (where [x,y] is and ordered pair.
    Also, let a and b be sets, Prove that there exists a set whose members are exactly the functions with domain a and codomain b

    Many thanks.
    Are you sure you've stated the question precisely? A relation is defined on a set. Domain and range are normally associated with a function.

    Could you possibly scan the exact problem in?
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  3. #3
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    Quote Originally Posted by oldguynewstudent View Post
    Are you sure you've stated the question precisely? A relation is defined on a set. Domain and range are normally associated with a function. Could you possibly scan the exact problem in?
    Actually, we say that a function is a relation with more restrictions.
    Therefore, we do define domains and codomains(images) for relations.
    That said I do have concerns with this question.
    Quote Originally Posted by KSM08 View Post
    Prove that there exists a set whose members are exactly the functions with domain a and codomain b
    In your text material what is expected when asked to prove existence of a set?
    Are you given a set of set axioms dealing with this?
    If so, we donít know what axioms you have to use.
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  4. #4
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    Quote Originally Posted by KSM08 View Post
    Show that domR={x : ∃y ([x, y] ∈ R)}, is a set
    Every x in dom(R) is a member of UUR. So the desired set is a subset of UUR, so we obtain the desired set by the axiom schema of separation. ['U' stands for the unary union operation.]

    Quote Originally Posted by KSM08 View Post
    Prove that there exists a set whose members are exactly the functions with domain a and codomain b
    I suppose that you've already proven the existence of Cartesian products. Then we observe that the desired set is a subset of P(aXb). So we obtain the desired set from the axiom schema of separation. ['P' stands for the power set operation and 'X' stands for the Cartesian product operation.]
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  5. #5
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    Quote Originally Posted by oldguynewstudent View Post
    Domain and range are normally associated with a function.
    Yes, but also some authors define 'dom' and 'range' on sets in general.

    dom(S) = {x | Ey <x y> e S}
    range(S) = {y Ex <x y> e S}

    whether S is a function, relation, or set of any kind whatsoever.
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  6. #6
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    Why be so stubborn, why not learn to post in standard symbols? You can use LaTeX tags.
    [tex] \text{dom}(S) = \left\{ {x|\left( {\exists y} \right)\left[ {\left( {x,y} \right) \in S} \right]} \right\}[/tex] gives  \text{dom}(S) = \left\{ {x|\left( {\exists y} \right) {\left( {x,y} \right) \in S} } \right\}.
    It is just so much easier to read standard notation.
    I can tell you,as an editor and a reviewer, that makes a difference in what gets published.
    Last edited by Plato; June 10th 2010 at 05:45 PM.
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