I need some help.. proof indexed family of set distributive law..

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I need some help.. proof indexed family of set distributive law..

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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hard to write because saying it says the same thing. but i think the way to do it is to show that each is contained in the other pick some element on the left, say \(x\) and show that it is in the right an vice versa if it is in the left, then either \(x\in B\) or \(x\in \cap A_{\alpha\in I}\) meaning \(\exists\beta\in I\) such that \(x\in A_\beta\)
then if \(x\in B\) it must be in \(A_{\alpha}\cup B\) for any \(\alpha\) and if \(a\notin B\) then by the proceeding it must be in some \(A_{\beta}\) oh crap i got my modifier wrong!!

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if \(x\in \cap A_{\alpha \in I}\) it is in ALL \(A_{\alpha}\)
so \(\forall \alpha \in I\) we know \(x\in A_{\alpha}\) and therefore \(x\in \cap (A_{\alpha \in I}\cup B)\)

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