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Simple set question

  1. Feb 16, 2005 #1
    I need to prove that [tex]-(-A)=A[/tex]. I guess it's the same as [tex]S-(S-A)=A[/tex], where [tex]S[/tex] is the space. So is it true, that if [tex]x \in S-(S-A)[/tex] then [tex]x \notin S-A[/tex]?

    - Kamataat
  2. jcsd
  3. Feb 16, 2005 #2
    I take it -A means the complement of A (with respect to some universe)? The following are equivalent (~ means "not"):

    x [itex]\in[/itex] -(-A)
    x [itex]\notin[/itex] -A
    ~(x [itex]\in[/itex] -A)
    ~(x [itex]\notin[/itex] A)
    ~(~(x [itex]\in[/itex] A))
    x [itex]\in[/itex] A

    That establishes the two inclusions -(-A) [itex]\subseteq[/itex] A and A [itex]\subseteq[/itex] -(-A).
  4. Feb 16, 2005 #3
    Thanks, Muzza, I get your proof. Still, why are the NOT steps neccessary? Why not this instead:

    [tex]x \in -(-A)[/tex]
    [tex]x \notin -A[/tex]
    [tex]x \in A[/tex]?

    If you go (in your post) from step #1 to step #2 directly, then why don't you go from #2 to #6 (e.g. skip #3, #4 and #5)? I mean, if from #1 follows #2, then doesn't #6 follow from #1 and #2 combined (w/o the intermediate steps)?

    - Kamataat
  5. Feb 16, 2005 #4
    *shrug* Do as you please :P
  6. Feb 16, 2005 #5
    ok, tnx

    - Kamataat
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