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How do I prove this? Sets

  1. Oct 4, 2011 #1
    1. The problem statement, all variables and given/known data


    For the sets A and B, prove that

    [tex]A \cap B \subseteq A \subseteq A \cup B[/tex]



    3. The attempt at a solution

    I am guessing I should look at only two of them first?

    [tex]A \subseteq A \cup B[/tex]

    What conditions do I need?
     
  2. jcsd
  3. Oct 4, 2011 #2

    SammyS

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    Definition of C ⊆ D ?

    Let x ∊ A⋂B, then ...
     
  4. Oct 4, 2011 #3
    let x be in A intersect B..what does that mean...
     
  5. Oct 4, 2011 #4
    C is a subset of D

    if x is in the intersection of A and B, then x belongs to both A and B
     
  6. Oct 4, 2011 #5
    so x is in A .... is x in the Union of A and B
     
  7. Oct 4, 2011 #6

    SammyS

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    ∴ x belongs to A .
     
  8. Oct 4, 2011 #7
    Ohhhh

    so for

    [tex]A \subseteq A \cup B[/tex]

    Same argument? i.e.

    [tex] A \cup B[/tex] for some element x, belongs to A or B and hence A also belongs to A? DOes the "or" say whether it can have elements in A or not? Is it a hasty conclusion?
     
  9. Oct 4, 2011 #8
    yeah your elements could be in either A or B, it helps to draw a picture
     
  10. Oct 4, 2011 #9

    SammyS

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    If x is in A, then x is in (A or B).
     
  11. Oct 5, 2011 #10
    I want to do this elegantly

    So

     
  12. Oct 5, 2011 #11
    I just want to ask, I don't need to show that [tex]A \cap B \subseteq A \cup B[/tex] right? Because this just follows from subset properties? Does this make a good proof?
     
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