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Power sets

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove [tex]P(A) \cup P(B) \subseteq P(A \cup B) [/tex]


    2. Relevant equations



    3. The attempt at a solution

    I started out by assuming that [tex] A = \left\{a\right\} [/tex] and [tex]B=\left\{b\right\}[/tex].

    So then [tex]P(A) \cup P(B) = \left\{\left\{a\right\},\left\{b\right\},null\right\}[/tex] and [tex]P(A \cup B) = \left\{\left\{a\right\},\left\{b\right\},\left\{a,b\right\},null\right\}[/tex]

    So I can conclude that [tex]P(A) \cup P(B) \subseteq P(A \cup B) [/tex]

    How does that sound?
     
  2. jcsd
  3. Oct 14, 2009 #2
    When you assume that A={a} and B={b}, you are assuming that A and B are both singleton sets. You want to prove the relation for any sets A and B.

    When proving one set is a subset of another, say X is a subset of Y, then you let x be in X and show x is in Y. So let [itex]x\in P(A)\cup P(B)[/itex], and then show [itex]x\in P(A\cup B)[/itex].
     
  4. Oct 14, 2009 #3
    Okay, so let [tex]x\in P(A)\cup P(B)[/tex].
    Then [tex]x\in P(A)[/tex] or [tex]x\in P(B)[/tex]... which means [tex]x\subseteq A[/tex] or [tex]x\subseteq B[/tex]?

    So then [tex]x\subseteq (A\cup B)[/tex]. Am I going in the right direction?
     
  5. Oct 14, 2009 #4

    Landau

    User Avatar
    Science Advisor

    Very good, the conclusion now follows.
     
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