## can you use induction on n cases (as opposed to infinity)?

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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.
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 Recognitions: Homework Help Science Advisor That is hideous to read. The basic idea is that E U (A_k+1 n E^c)=E U A_k+1. Right? You can certainly use induction on a finite set of premises, no problem with that. It looks ok to me. Cleaning up the presentation certainly wouldn't hurt. Using TeX wouldn't hurt either. But I think you've got one way to do it.

## can you use induction on n cases (as opposed to infinity)?

The purpose of induction is to show that if a statement is true for some value k, it has to be true for k+1.

It's up to you how far you want to extend your conclusion, so it's perfectly fine to use it on a finite set.

alright, sorry i was a bit lazy on the latex, i didn't think it would be that bad originally, and i haven't used latex in a while.

i've attached a pdf. how does that look?
Attached Files
 borel.pdf (69.6 KB, 9 views)

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