Set theoretic problem, lim sup and lim inf

[sunjin09]

Homework Statement

Given a sequence of sets {An}, n=1,2,...,∞, the lim sup $S=\cap_{n=1}^\infty\cup^\infty_{k=n}A_k$, and the lim inf $I=\cup_{n=1}^\infty\cap^\infty_{k=n}A_k$, obviously $I\subset S$, find an expression for the set difference $S-I$

Homework Equations

$(A \cup B) \cap C = (A \cap C)\cup(B\cap C)\,$
$(A \cap B) \cup C = (A \cup C)\cap(B\cup C)\,$

The Attempt at a Solution

I don't know how to use the set algebra identities to simplify, I need to write S-I in terms of set differences of An. Are there any identities involving infinite union/intersection that might be of help? Thank you.

 

[mighty2000]

Greetings, fellow scientist!

To find an expression for the set difference S-I, we can use the following identity:

(A \cup B) \setminus C = (A \setminus C) \cup (B \setminus C)

Applying this to our problem, we have:

S-I = (S \setminus I) \cup (I \setminus S)

Using the given expressions for S and I, we have:

S-I = (\cap_{n=1}^\infty\cup^\infty_{k=n}A_k) \setminus (\cup_{n=1}^\infty\cap^\infty_{k=n}A_k)

Now, using the definition of set difference, we have:

S-I = \{x \mid x \in (\cap_{n=1}^\infty\cup^\infty_{k=n}A_k) \text{ and } x \notin (\cup_{n=1}^\infty\cap^\infty_{k=n}A_k)\}

Simplifying the above expression, we get:

S-I = \{x \mid x \in A_n \text{ for some } n \text{ and } x \notin A_m \text{ for all } m \geq n\}

In other words, the elements in S-I are those that are in at least one of the sets An, but not in any of the sets beyond An.

I hope this helps! Let me know if you have any further questions.