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bennyska
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Homework Statement
this is probably a dumb question, but I'm doing this proof where i have to show two sets are equal, where each set is a union from 1 to n sets. this is pretty easy to show with induction, i think, but I'm used to using induction when i have an infinite amount of things, so I'm not sure I'm allowed to use induction. any thoughts?
specifically, it goes like this:
Suppose that A_1, ..., An are Borel sets, that is they belong to ß. Define
the following sets: B_1 = A_1, B_n = A_n ∩ (A_1∪ ... ∪ A_n-–1)^c (^c is complement), and let S equal the universal set. Show that
U_i=1 to n A_i = U_i=1 to n B_i.
Homework Equations
The Attempt at a Solution
U_1 to 1 A_i = A_1 = U_1 to 1 B_i = B_1. So we have a base case. So assume it's true for n=k. Then we have that U_i=1 to k A_i = U_i=1 to k B_i.
Then we have that U_i to k B_i U B_k+1 = U_i to k A_i U (A_k+1 ∩ (A_1∪ ... ∪ A_k)^c
=U_i to k A_i U (A_k+1 ∩ A_1^c ∩ A_2^c...∩A_k^c)...
Let A_1^c ∩ A_2^c...∩A_k^c = D, and let U_i to k A_i = E
Then we have U_i to k B_i U B_k+1 = E U (A_k+1 ∩ D)
= (E U D) ∩ (E U A_k+1) = S ∩ (U_i to k A_i U A_k+1) = U_i=1 to k+1 A_i.
god that looks hideous. hopefully it makes sense. any comments would be appreciated.