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Is the following set equal to the empty set?

  1. Apr 13, 2009 #1
    Is the following set equal to the empty set??

    A={x:[tex] x\in A\Longrightarrow y\in A ,x\neq y[/tex]},if yes prove it ,if not disproved it
     
  2. jcsd
  3. Apr 13, 2009 #2
    Re: sets

    This doesn't make sense. For one thing any thing that is not in A will be in A.
     
  4. Apr 13, 2009 #3
    Re: sets

    A Is a set of x elements in a such way that if x belongs to A THEN ANY y different from x belongs to A.

    Doesn't that make sense??
     
  5. Apr 13, 2009 #4
    Re: sets

    Well, if A existed, it would be the universal set (because of what Focus said). But there is no universal set. So A is not a set.
     
  6. Apr 13, 2009 #5

    CRGreathouse

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    Re: sets

    This is just
    [tex]\forall x\;\; x\in A\Longleftrightarrow \left(x\in A\Rightarrow\forall y\neq x\;\; y\in A\right)[/tex]
    which is
    [tex]\forall x\;\; x\in A\Longrightarrow \left(x\in A\Rightarrow\forall y\neq x\;\; y\in A\right)[/tex]
    [tex]\left(x\in A\Rightarrow\forall y\neq x\;\; y\in A\right)\Longrightarrow\forall x\;\; x\in A[/tex]
    which is
    [tex]\forall x\;\; x\in A\Longrightarrow \left(\forall y\neq x\;\; y\in A\right)[/tex]
    [tex]\forall x\;\; \left(x\in A\Rightarrow\forall y\neq x\;\; y\in A\right)\Longrightarrow x\in A[/tex]
    which is
    [tex]\forall x\;\; x\in A\Longrightarrow \left(\forall y\neq x\;\; y\in A\right)[/tex]
    [tex]\forall x\;\; x\in A[/tex]
    which is
    [tex]\forall x\forall y\neq x\;\; y\in A[/tex]
    [tex]\forall x\;\; x\in A[/tex]
    which is
    [tex]A=\mathcal{U}[/tex]

    So if you have a universal set, A is it; if not, the definition is ill-defined.
    Edit: What Preno said.
     
  7. Apr 13, 2009 #6
    Re: sets

    .

    LETS take it line by line:

    1st line you have written: [tex]\forall x\;\; x\in A\Longleftrightarrow \left(x\in A\Rightarrow\forall y\neq x\;\; y\in A\right)[/tex].

    DO you actually mean:[tex]x\in A\Longleftrightarrow(x\in A\Longrightarrow\forall y(y\neq x\wedge y\in A))[/tex].

    If yes, how did you get that?
     
  8. Apr 13, 2009 #7

    CRGreathouse

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    Re: sets

    I'm just expanding the definition of set-builder notation.
     
  9. Apr 14, 2009 #8
    Re: sets

    expanding set builder notation is:

    [tex] x\in A\Longleftrightarrow[(x\in A\Longrightarrow y\in A)\wedge x\neq y][/tex]

    Now how from the above you get :


    [tex]\forall x\;\; x\in A\Longleftrightarrow \left(x\in A\Rightarrow\forall y\neq x\;\; y\in A\right)[/tex].
     
  10. Apr 14, 2009 #9

    CRGreathouse

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    Re: sets

    Wait, you really meant
    [tex](x\in A\Longrightarrow y\in A)\wedge x\neq y[/tex]
    in your original formulation when you wrote
    [tex]x\in A\Longrightarrow y\in A ,x\neq y[/tex]?

    That's very different!
     
  11. Apr 14, 2009 #10
    Re: sets

    yes

    A= { [tex]x: (x\in A\Longrightarrow y\in A)\wedge x\neq y[/tex]}

    sorry
     
  12. Apr 14, 2009 #11

    CRGreathouse

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    Re: sets

    So your definition is given in terms of the unbound variable y? I was assuming in my translation that you intended for the definition to be a sentence.
     
  13. Apr 14, 2009 #12
    Re: sets

    the whole sentence
     
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