# Math Help - Power set on intersection

1. ## Power set on intersection

Proof the proposition.

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?
which is all K ∈ I, X∈ Sk
which is all K ∈ J, X∈ Sk

Some hints to get me start the question would be helpful. Thank you.

2. ## Re: Power set on intersection

Originally Posted by kcyw0515
Proof the proposition.

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?
which is all K ∈ I, X∈ Sk
which 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?

3. ## Re: Power set on intersection

Originally Posted by Plato
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.

I know that J ⊆ I, but in order to proof .

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

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,
$$J\subseteq I\implies \bigcap_{k\in I}S_k\subseteq\bigcap_{k\in J}S_k.$$