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Math Help - Relations Proof

  1. #1
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    Relations Proof

    Let A be a nonempty subset and let B be a fixed subset of A. A relation R defined on the set of subsets of A is defined by XRY if (X intersects B) = (Y intersects B)

    (a) Prove that R is an equivalence relation.
    (b) Let A = {1,2,3,4,5} and B={1,3,5}. For X = {1,2,3}, find [X]

    Help please? I'm working on a review before finals and I haven't seen a problem like this in months, and I can't remember how to do it. So for (a) I remember you show two cases, first that X is a subset of Y, then that Y is a subset of X, but I don't remember how to do that. I'm completely lost on (b).

    Thanks in advance!
    Last edited by IHateProofs; April 25th 2014 at 07:06 AM.
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  2. #2
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    Re: Relations Proof

    Quote Originally Posted by IHateProofs View Post
    Let A be a nonempty subset and let B be a fixed subset of A. A relation R defined on the set of subsets of A is defined by XRY if (X intersects B) = (Y intersects B)
    (a) Prove that R is an equivalence relation.
    (b) Let A = {1,2,3,4,5} and B={1,3,5}. For X = {1,2,3}, find [X]
    a) Show that the relation is reflexive, symmetric, and transitive. Show us what you have done.
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    Re: Relations Proof

    So after looking through months of old notes I think I finally found how to do (a). This is what I have.

    Proof: We will prove this by showing that R possesses the reflexive, symmetric, and transitive properties. First we will prove that R is reflexive. Observe X ∩ B = X ∩ B. Therefore XRX and R is reflexive. Next assume XRY. Then X ∩ B = Y ∩ B. Hence Y ∩ B = X ∩ B. Therefore YRX and R is symmetric. Finally, assume XRY and YRZ. Then X ∩ B = Y ∩ B, and Y ∩ B = Z ∩ B. Hence X ∩ B = Z ∩ B. Therefore XRZ, and so R is transitive. Since R possesses the reflexive, symmetric, and transitive properties, it follows that R is an equivalence relation.

    But I still can't figure out (b). I can't find anything in my old assignments about the equivalence class of a subset. If I had to guess, I'd say the answer is the super set of X ∩ B = (1,3). So [X]= {(1,3), X, (1,3,4), B, (1,2,3,4), (1,2,3,5), (1,3,4,5), A}. Is that right?
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  4. #4
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    Re: Relations Proof

    Quote Originally Posted by IHateProofs View Post
    So after looking through months of old notes I think I finally found how to do (a). This is what I have.

    Proof: We will prove this by showing that R possesses the reflexive, symmetric, and transitive properties. First we will prove that R is reflexive. Observe X ∩ B = X ∩ B. Therefore XRX and R is reflexive. Next assume XRY. Then X ∩ B = Y ∩ B. Hence Y ∩ B = X ∩ B. Therefore YRX and R is symmetric. Finally, assume XRY and YRZ. Then X ∩ B = Y ∩ B, and Y ∩ B = Z ∩ B. Hence X ∩ B = Z ∩ B. Therefore XRZ, and so R is transitive. Since R possesses the reflexive, symmetric, and transitive properties, it follows that R is an equivalence relation.

    But I still can't figure out (b). I can't find anything in my old assignments about the equivalence class of a subset. If I had to guess, I'd say the answer is the super set of X ∩ B = (1,3). So [X]= {(1,3), X, (1,3,4), B, (1,2,3,4), (1,2,3,5), (1,3,4,5), A}. Is that right?
    Yes you did the (a) part correctly.

    In the part (b) your notation is off. It should be $X\cap B=\{1,3\}$ NOT $(1,3)$.
    This a relation on subsets not ordered pairs.
    But because $A\cap B=B~\&~X\cap B\ne B$ then $A\notin [X]$.

    So redo part (b).
    Last edited by Plato; April 26th 2014 at 01:09 PM.
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  5. #5
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    Re: Relations Proof

    so...
    [X]= {{1,3}, X, {1,3,4}, {1,2,3,4}}
    yes?
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  6. #6
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    Re: Relations Proof

    Quote Originally Posted by IHateProofs View Post
    so...
    [X]= {{1,3}, X, {1,3,4}, {1,2,3,4}}
    yes?
    Yes.
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    Re: Relations Proof

    Awesome thanks!
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