## cinar 2 years ago I need some help.. proof indexed family of set distributive law..

1. cinar

2. satellite73

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 $$x\in A_\beta$$

3. satellite73

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!!

4. satellite73

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

5. satellite73

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