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Math Help - Partition of a set

  1. #1
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    Partition of a set

    The question is to determine whether \mathcal{A} is a partition of the set A.

    A = \mathbb{R} and \mathcal{A} = \{S_y : y \epsilon \mathbb{R}\}, where S_y = \{x \epsilon \mathbb{R} : |x| = |y|\}.

    I understand what it means to be a partition, but I'm not sure what \mathcal{A}, or particularly, S_y means.

    Does this mean \mathcal{A} = \{ \{1, 1\}, \{1.1, 1.1\}, \{1.2, 1.2\}, ..., \{-1, 1\}, \{-1.1, 1.1\}, \{-1.2, 1.2\}, ... \{1, -1\}, \{1.1, -1.1\}, ... \{-1, -1\}, ... \{-5.4, 5.4\} \}? Where I just find any real number pair so that their absolute values are equal?
    Last edited by deezy; October 25th 2012 at 05:46 PM.
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  2. #2
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    Re: Partition of a set

    S_x = \{ x, -x \} \text{ for } x \in \mathbb{R}, x \ne 0. \ S_0 = \{ 0 \}. \  \mathcal{A} = \{ S_y : y \in \mathbb{R} \}.

    \text{Since } S_x = S_{-x} \text{ for } x \ne 0, \text{ have that }

    \mathcal{A} = \{ S_y : y \in \mathbb{R} \} = \{ S_y : y \in \mathbb{R}, y \ge 0 \}.

    Are those sets in \mathcal{A} disjoint? Non-empty? Does their union equal the whole set (in this case, \mathbb{R})? If so, then \mathcal{A} is a partition of \mathbb{R}.

    --------------------------
    The only potentially confusing thing here is that the original definition of \mathcal{A} had redundancies - two different names for the same set - and so that might seem to make it not a partition. However, a set with redundant (even if differently named) elements is the same as the set with unique elements.

    i.e. If a = -2, b = -2, then U = {a, b} = { -2, -2 } = { -2 } = { a }.

    Likewise: If A = {5, -2}, B = {5, -2}, then V = {A, B} = { {5,-2}, {5,-2} } = { {5,-2} } = { A }.
    Last edited by johnsomeone; October 25th 2012 at 07:47 PM.
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