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Math Help - Power set on intersection

  1. #1
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    Power set on intersection

    Proof the proposition.
    Power set on intersection-untitled.png

    For this question. I have no idea how to start.
    The P(I) just messes me up.
    Since J belongs in the set P(I), shouldn't the statement be instead of the top one?
    Power set on intersection-untitled.pngwhich is all K ∈ I, X∈ Sk
    Power set on intersection-untitled.pngwhich is all K ∈ J, X∈ Sk



    Some hints to get me start the question would be helpful. Thank you.
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  2. #2
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    Re: Power set on intersection

    Quote Originally Posted by kcyw0515 View Post
    Proof the proposition.
    Click image for larger version. 

Name:	Untitled.png 
Views:	1 
Size:	4.4 KB 
ID:	30523

    For this question. I have no idea how to start.
    The P(I) just messes me up.
    Since J belongs in the set P(I), shouldn't the statement be instead of the top one?
    Click image for larger version. 

Name:	Untitled.png 
Views:	23 
Size:	1.6 KB 
ID:	30524which is all K ∈ I, X∈ Sk
    Click image for larger version. 

Name:	Untitled.png 
Views:	23 
Size:	1.7 KB 
ID:	30525which is all K ∈ J, X∈ Sk
    Some hints to get me start the question would be helpful. Thank you.
    To say that $J\in\mathcal{P}(I)$ is to simply say $J\subseteq I$.

    Surely that means $\bigcap\limits_{k \in I} {{S_k}} \subseteq \bigcap\limits_{k \in J} {{S_k}} $ NO?
    Thanks from kcyw0515
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  3. #3
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    Re: Power set on intersection

    Quote Originally Posted by Plato View Post
    To say that $J\in\mathcal{P}(I)$ is to simply say $J\subseteq I$.

    Surely that means $\bigcap\limits_{k \in I} {{S_k}} \subseteq \bigcap\limits_{k \in J} {{S_k}} $ NO?
    First of all, Thank for the help

    I get what you are saying, but i'm kind of confused.
    I'm not trying to sound stupid, just that my prof is really awful.

    From the textbook.

    Power set on intersection-untitled2.png

    Power set on intersection-untitled.png

    I know that J ⊆ I, but in order to proof Power set on intersection-untitled.png.

    Don't i need to show I ⊆ J?
    And how would i show that?


    Thank for the advise.
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  4. #4
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    Re: Power set on intersection

    Imagine that $I$ and $J$ are groups of people and $J\subseteq I$. Also, for $k\in I$, let $S_k$ be the set of days where person $k$ is free to come to a movie night. To find the set of days that work for all people in $I$, we take $\bigcap_{k\in I}S_k$, and similarly for $J$. Now, clearly, the more people, the harder is to find a night that suits everyone, i.e., the smaller the intersection of $S_k$ is. Formally,
    $$J\subseteq I\implies \bigcap_{k\in I}S_k\subseteq\bigcap_{k\in J}S_k.$$
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