Proof of A U (intersect of B1 to Bn) = intersect of A U B1 to Bn

[rhyno89]

Homework Statement

prove the following:A U (intersect from i = 1 to n of Bi) = intersect of i from 1 to n of (A U B)

Homework Equations

The Attempt at a Solution

i started off trying to rewrite it and make it a bit more readable
A U (Bi and Bi+1) = Bi+1 and (A U Bi)
then i was going to expand them and write the outcomes and show that they have the same as the proof

for the first part i got: AA ABi ABi+1 and both Bs

my problem is that i think i messed up the right hand side when rewriting it because that way the most i can get is 2 outcomes

any thoughts?

 

[ystael]

This is really simple set theory, not so much probability theory. If you get lost doing something like this, the easiest thing to do is break it down to the definitions and follow your nose.

For sets (events) $$C$$ and $$D$$, the equation $$C = D$$ means "both $$C \subset D$$ and $$C \supset D$$". So, to prove an equation between sets, prove inclusion in both directions.

The inclusion $$C \subset D$$ means "if $$x \in C$$, then $$x \in D$$". So, to prove an inclusion between sets, prove that every point in the first set is contained in the second one.

So, you want to prove $$A \cup \bigcap_{i=1}^n B_i = \bigcap_{i=1}^n (A \cup B_i)$$. Break it down into proving the inclusions in each direction. So suppose $$x \in A \cup \bigcap_{i=1}^n B_i$$; you want to prove that $$x \in \bigcap_{i=1}^n (A \cup B_i)$$. What does it mean for $$x$$ to belong to a union? To an intersection? Follow the definitions, one step at a time. Then prove the reverse inclusion.