Question

I need some help..

proof indexed family of set distributive law..

Answer 1

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 \(x\in A_\beta\)

Answer 2

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!!

Answer 3

if \(x\in \cap A_{\alpha \in I}\) it is in ALL \(A_{\alpha}\)

Answer 4

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