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Math Help - Power Sets Proof

  1. #1
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    Power Sets Proof

    Prove the following:

    Let  A and  B be sets. Then  A \subseteq B iff  \mathcal {P}(A) \subseteq \mathcal{P}(B) .

    Here is what I am thinking, but I'm pretty sure there is some serious problems:

    Let  A and  B be sets.
    First Direction: Assume  A \subseteq B . Let  x \subseteq A . Since  x \subseteq A, x \in \mathcal {P}(A) . Since  A \subseteq B, x \subseteq B . Because  x \subseteq B, x \in \mathcal{P}(B) . Since  A \subseteq B, \mathcal {P}(A) \subseteq \mathcal{P}(B) .

    Second Direction: Assume  \mathcal{P}(A) \subseteq \mathcal{P}(B) . Let  x \in \mathcal{P}(A) . Since  x \in \mathcal {P}(A), x \subseteq A . Since \mathcal{P}(A) \subseteq \mathcal{P}(B), x \in \mathcal{P}(B) . Since  x \in \mathcal{P}(B), x \subseteq B . Since x \subseteq A and  x \subseteq B, A \subseteq B.


    Need some help please. Thanks.
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  2. #2
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    You have the symbolic notation confused.
    If A \subseteq B then prove that \wp \left( A \right) \subseteq \wp \left( B \right).
    Start by picking a point G \in \wp \left( A \right), by definition G \subseteq A.
    Now can you finish be concluding that G \in \wp \left( B \right)?


    Next start with \wp \left( A \right) \subseteq \wp \left( B \right) pick x \in A.
    We know that \left\{ x \right\} \subseteq A. Can you finish by showing <br />
x \in B?
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  3. #3
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    Okay, let's see what I can do here.

    First way.
    Assume  A \subseteq B .
    Let  G \in \mathcal{P}(A) .
    Thus  G \subseteq A.
    Since  A \subseteq B, G \subseteq B.
    Because  G \subseteq B, G \in \mathcal{P}(B).
    Hence, if  A \subseteq B , then  \mathcal{P}(A) \subseteq \mathcal{P}(B).

    Second way.
    Assume  \mathcal{P}(A) \subseteq \mathcal{P}(B) .
    Let  x \in A.
    Thus, {x}  \subseteq A.
    Since {x}   \subseteq A, x \in \mathcal{P}(A).
    Since  \mathcal{P}(A) \subseteq \mathcal{P}(B), x \in \mathcal{P}(B) .
    Since  x \in \mathcal{P}(B), {x}  \subseteq B .
    Because {x}  \subseteq B, x \in B.
    Hence, if  \mathcal{P}(A) \subseteq \mathcal{P}(B), A \subseteq B.

    Is this okay? Or did I do something wrong again? I'm pretty confused.
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  4. #4
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    Since {x}   \subseteq A, \color{red}{\{x\}} \in \mathcal{P}(A).
    Since  \mathcal{P}(A) \subseteq \mathcal{P}(B), \color{red}{\{x\}} \in \mathcal{P}(B) .
    Since  \color{red}{\{x\}} \in \mathcal{P}(B), {x}  \subseteq B .
    Because {x}  \subseteq B, x \in B.
    Hence, if  \mathcal{P}(A) \subseteq \mathcal{P}(B), A \subseteq B.
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  5. #5
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    Thanks, that's what I had written on my paper.

    I just screwed up when I typed it out.

    You've been a great help.

    Thanks so much.
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