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Math Help - theorem about family of sets

  1. #1
    Member
    Joined
    Oct 2010
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    Mumbai, India
    Posts
    203

    theorem about family of sets

    Here's the theorem I am trying prove

    Suppose A is a set, and for every family of sets \mathcal{F}
    we have

    \left[\bigcup \mathcal{F} = A\right]\Rightarrow A \in \mathcal{F}

    so prove that A has exactly one element. Now there are two parts here. One is the existence part and second is uniqueness part. I have proved the
    existence part using method of contradiction. I chose \mathcal{F}=\varnothing as a particular F to show the contradiction.

    Now coming to the uniqueness part, the given is

    Given ---->

    \forall \mathcal{F} \left[(\bigcup \mathcal{F} = A )\Rightarrow A \in \mathcal{F}\right]

    and the Goal is

    \forall y \forall z \left[(y \in A)\wedge (z \in A)\Rightarrow (y=z)\right]

    I will assume the negation of the goal and try to find the contradiction. Hence the
    givens are

    \forall \mathcal{F} \left[(\bigcup \mathcal{F} = A )\Rightarrow A \in \mathcal{F}\right]

    (y \in A)\wedge (z \in A)

    y \neq z

    and the goal now is contradiction. Now here I am having some difficulty,
    how do I choose particular \mathcal{F} ? Since y \in A
    and z \in A we know that

    \{y,z\} \subseteq A

    But there could be more elements in A. Is there any way I can consider different
    cases which will exhaust the possibilities ?

    please comment.
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  2. #2
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: theorem about family of sets

    The method of contradiction is fine for existence, but probably not best for uniqueness. Suppose towards a contradiction that A=\emptyset. Then \bigcup\emptyset=\emptyset=A and hence A\in\emptyset, a contradiction. So there is a\in A. Let \mathcal{F}=\{\{a\},A\setminus\{a\}\}. Then \bigcup\mathcal{F}=A and hence A\in\mathcal{F}. Since A\neq A\setminus\{a\} it follows that A=\{a\}.
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  3. #3
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: theorem about family of sets

    hatsoff

    thanks , that is just beautiful...

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