Big Unions/Intersections

Question

I have a quick and simple question in regards to what a "Big Union/Intersection" even is. I saw this term in my text book, but when I did some research there is nothing on it. Hopefully this article will serve as a resource for future researchers.

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I am not looking for answers, I am looking at for the direction I need to research and learn this stuff so That I can solve those problems myself.

These are the problems which yielded no results when I tried to research them, ordered by my lack of understanding.

1a,3,2

Answer 1

I believe it just means the union/intersection of a large number of sets.
For example, $$ \bigcup_{i=1}^{n} A_{i} = A_{1} \cup A_{2} \cup \cdots \cup A_{n-1} \cup A_{n}$$ $$ \bigcap_{i=1}^{n} A_{i} = A_{1} \cap A_{2} \cap \cdots \cap A_{n-1} \cap A_{n}$$

Answer 2

For a finite union of sets $ S_1, S_2, S_3, \dots , S_n$, one often writes $ S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n $ or $ {\displaystyle \bigcup _{i=1}^{n}S_{i}}$. In the case that the index set I is the set of natural numbers, one uses a notation ${ \displaystyle \bigcup_{i=1}^{\infty} A_{i}}$ analogous to that of the infinite series. We can similarly explain the notation for that of the intersection of the sets. Hope it helps.

Answer 3

There are general, simple definitions for the "big union" and "big intersection":

Let $\Lambda$ be an index set and $\mathscr{C} = \{A_{\lambda}\}_{\lambda \in \Lambda}$ be a collection of sets $A_{\lambda}$. We define

$\bigcup_\limits{\lambda \in \Lambda}A_{\lambda} = \{ a: \exists \ \lambda \in \Lambda : a\in A_{\lambda} \}$

$\bigcap_\limits{\lambda \in \Lambda}A_{\lambda} = \{ a: \forall \ \lambda \in \Lambda : a\in A_{\lambda} \}$

Observe that these definitions coincide with the intuitive notion of union and intersection of a finite amount of sets, besides they extent for an infinite (denumerable or not) collection of sets.