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Elementary proof check/help

  1. Sep 28, 2012 #1
    1. The problem statement, all variables and given/known data
    For all sets A, B, and C, prove or provide a counterexample the following statements.
    (A [itex]\setminus[/itex] B) [itex]\cap[/itex] (C [itex]\setminus[/itex] B) = A [itex]\setminus[/itex] (B [itex]\cup[/itex] C).


    2. Relevant equations
    N/A


    3. The attempt at a solution
    I went ahead and said it was false, and provided a counter example. I'm new to this and just wanna make sure my thought process was correct and the statement is indeed false.
    Counterexample:
    Let A = {1,2,3}, B = {2,3}, and C = {1,2}.
    A [itex]\setminus[/itex] B = {1} and C [itex]\setminus[/itex] B = {1}. Then (A [itex]\setminus[/itex] B) [itex]\cap[/itex] (C [itex]\setminus[/itex] B) = {1}.
    B [itex]\cup[/itex] C = {1,2,3}. Then A [itex]\setminus[/itex] (B [itex]\cup[/itex] C) = {0}. Since {1} ≠ {0}, the statement is false.

    It's more the last part, i.e. "Then A [itex]\setminus[/itex] (B [itex]\cup[/itex] C) = {0}" I want to make sure is correct. Thanks for your help!
     
  2. jcsd
  3. Sep 28, 2012 #2

    jbunniii

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    This is almost right. Your notation {0} is incorrect. That refers to a set containing one element, the number 0. What you want is [itex]\emptyset[/itex], the empty set.

    Aside from that your counterexample looks fine.
     
  4. Sep 29, 2012 #3
    I meant the empty set, but I guess 0 would be element and not empty. Thanks for telling the difference, I won't make that same mistake again.
     
  5. Sep 29, 2012 #4

    HallsofIvy

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    If you want to use "regular" set notation, rather than ∅, it would be "{}", not "{0}".
     
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