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Help with proving a set

  1. Feb 21, 2016 #1
    1. The problem statement, all variables and given/known data
    Number 1 20160221_161544.jpg

    2. Relevant equations
    I know I should use a Venn diagram.
    3. The attempt at a solution
    The statement says that x intersects y so therefore the statement equals x minus y. This Is my attempt at the solution, if you subtract y from the venn diagram you get this partially eaten cookie shape... So would that be enough to prove a it? My explanation would be that since what is left of x is clearly not a full set and some where y intersects x. Untitled.png
     
    Last edited: Feb 21, 2016
  2. jcsd
  3. Feb 21, 2016 #2

    andrewkirk

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    Science Advisor
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    A Venn diagram is not a proof.
    I think they suggest you look at the Venn diagram to get an idea of why it's true, which should point you towards a formal proof, not for the diagram to serve as a proof.
    An easy way to prove this is to write both sides in set-builder notation. Once you've done that, it's pretty easy to transform one side into the other using basic logical operations.
     
  4. Feb 21, 2016 #3
    X ∩ Y(line above the Y) = {ℤ∈ℤ | ℤ∈x ∨ ℤ ∈Y} would this be the correct set builder notation.
     
  5. Feb 21, 2016 #4

    Mark44

    Staff: Mentor

    No. For one thing, ℤ∈ℤ doesn't make any sense, because you haven't said what ℤ is, and a set is not an element of itself.

    For another thing, ℤ∈x ∨ ℤ ∈Y, use lower case for elements of a set, and upper case for the sets themselves. I would write what you have as {##z \in U | z \in X ∨ z \in Y##}, where U is the universal set. But that isn't correct either. The symbol ∨ is the logical or -- for sets use U for union.

    As mentioned, my revision of what you wrote is still wrong. It represents all elements that belong to X, together with all elements that belong to Y. That isn't what you want.
     
  6. Feb 21, 2016 #5
    So for example I set A= X-Y and using the definition of set difference. A∈x and A∉Y. The definition of complement x ∉y implies x ∉Y(line above it). Then the definition of intersection makes it x∈X∩Y(line above it).
    Then I would just to do the same thing with the other part correct?
     
  7. Feb 21, 2016 #6

    Mark44

    Staff: Mentor

    There's no need to bring another set into the mix.
    This makes no sense. As you have defined A (which you really don't need), it's a set. Use x (lowercase) for set elements and X (uppercase) for set names.
    You want to show that if ##x \in X \cap \overline{Y}##, then ##x \in X - Y##. That shows that ##X \cap \overline{Y} \subset X - Y##. Then you need to go the other way: if ##x \in X - Y##, then ##x \in X \cap \overline{Y}##, which will show that ##X - Y \subset X \cap \overline{Y}##. Together, these two proofs show that the two sets are equal.

    I used LaTeX for my set notation, with ## at the beginning of each, and the same at the end.
    \overline{Y} to make ##\overline{Y}##
    \subset to make ##\subset##
    \cap to make ##\cap##
     
  8. Feb 21, 2016 #7
    That is a lot simpler way then what I did. And I'm sorry for my syntax errors
     
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