Hi everyone, I have a question about some properties of limit of sequence of sets. Here we go:
Let and be two sequences of sets such that they both converge. That is, and . Can we conclude any of the following?
iv. where denotes symmetric difference
Thanks for your help.
Thanks for your interest in my question. Let be a sequence of sets in the class . Then, we define the lower limit and the upper limit of this sequence as and respectively. We say that exists if and is equal to the lower and the upper limits.
If A is a set, let in(x, A) = 1 if x is in A and in(x, A) = 0 otherwise. Then lim (A_n) exists iff for every x, the limit of in(x, A_n) exists, i.e., in(x, A_n) eventually stabilizes. Also, x is in lim(A_n) iff in(x, A_n) stabilizes at 1. It is easy to express in(x, A * B) through in(x, A) and in(x, B) where * is each of the four set operations in the question. Using these facts, it is possible to show that the limits in the left-hand sides exist and the equalities hold.