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Can you check this proof please: sets?

  1. Feb 11, 2009 #1
    1. The problem statement, all variables and given/known data

    A,B,C,D are sets.

    Prove that if C is contained in A and D is contained in B, then C∩ D is contained in A∩ B.

    2. Relevant equations



    3. The attempt at a solution
    Let x be any element.

    Then There exists (x that belongs to C∩D) and (x does not belong to A∩ B)

    So x belongs to C and x belongs to D

    If x belongs to C, since C is contained in A, then x belongs to A.
    If x belongs to D, since D is contained in B, then x belongs to B.
    So x belongs to A intersect B, a contradiction.

    Then the original statement is true.

    Is it okay?
     
  2. jcsd
  3. Feb 11, 2009 #2

    Tom Mattson

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    That needn't be the case at all. Let [itex]A=B=C=D=\mathbb{R}[/itex].
     
  4. Feb 11, 2009 #3
    Well, then can I say to assume that because I am trying to do a proof by contradiction.

    I'm confused by what you are saying....I should say A=B=C=D=R and then what? I'm not sure I follow...is the rest okay?
     
  5. Feb 11, 2009 #4

    Office_Shredder

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    Don't do proof by contradiction, it just muddles things. Everything else was fine.

    Hence all x in C intersect D are in A intersect B, and C intersect D is a subset of A intersect B. It's much cleaner that way
     
  6. Feb 11, 2009 #5

    Tom Mattson

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    I was giving you a counterexample to demonstrate that your opening statement is false. If you let all 4 sets equal the real numbers then it is not the case that that there exists an [itex]x\in C\cap D[/itex] with [itex]x\notin A\cap B[/itex].
     
  7. Feb 11, 2009 #6

    Tom Mattson

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    I agree with Office Shredder that you should forget about proof by contradiction here, but I don't agree that this all by itself is fine. You should include a line that says that [itex]x\in C\cap D[/itex]. After all, you're supposed to show that [itex](C \cap D) \subset (A \cap B)[/itex]. Those two sets should be connected by your argument.
     
  8. Feb 11, 2009 #7
    thanks guys...I actually I did include such a line in my formal write-up so thanks
     
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