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Are there sets that are not partitionable in certain ways?

  1. May 20, 2005 #1

    quasar987

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    Are there sets that are not partitionable in certain ways? For exemple, can I partition [itex]\mathbb{R}[/itex] into a collection of singletons?

    Can I partition [itex]\mathbb{R}^2[/itex] into a collection of lines of slope 2?

    If so, how would you write each of those partitions?

    Thx.
     
  2. jcsd
  3. May 20, 2005 #2
    For the first one: for every [tex]x \in \mathbb{R}[/tex] let [tex]U_x = \{x\}[/tex]. Then [tex]\cup_{x \in \mathbb{R}} U_x = \mathbb{R}[/tex] and obviously all [tex]U_x[/tex] are disjoint from one another, and are singletons.

    For the second: for every [tex]a \in \mathbb{R}[/tex] let [tex]U_a = \{(x, y) | y = 2x + a\}[/tex] (I wish I knew LaTeX better).

    Then take [tex](x_0, y_0) \in \mathbb{R}^2[/tex] and notice that [tex](x_0, y_0) \in U_{y_0 - 2x_0}[/tex]. Thus [tex]\cup_{a \in \mathbb{R}} U_a = \mathbb{R}^2[/tex]. A simple calculation will also reveal that [tex]U_a[/tex] and [tex] U_b[/tex] are either equal or disjoint (for any real a, b).
     
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