# Big Unions/Intersections

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.

Here is a screenshot of the questions

http://prntscr.com/ddmvyd

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

• Commented Nov 30, 2016 at 14:22
• In general, if $\oplus$ is an associative binary operation, then $\bigoplus_{x \in A} x$ denotes the $\oplus$ of all the elements in $A$; if $A$ is not finite, then $\oplus$ may need to be extended in some limiting way (e.g., as addition is). Commented Nov 30, 2016 at 21:22

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.

• The index is not necessary (though sometimes it is useful). If $\mathscr C$ is a collection of sets then $\bigcup \mathscr C := \{a \mid \exists A \in \mathscr C\ :\ a \in A\}, \bigcap \mathscr C := \{a \mid \forall A \in \mathscr C, a \in A\}$, etc. However, these are generally just called "union" and "intersection", no matter how big $\mathscr C$ is. The "big" adjective is just someone's personal description. Commented Nov 30, 2016 at 16:42
• Thanks for adding that! I just used the word "big" because Nick Lim used it in his/her question, so I was making reference to that. Commented Nov 30, 2016 at 18:45
• I figured that. I pointed it out for the sake of Nick Lim, to make sure he realized that the "big" was not indicating anything special. Commented Nov 30, 2016 at 21:17
• @PaulSinclair: As an aside, $\bigcup \mathscr{C} = \bigcup_{S \in \mathscr{C}} S$, and similarly for intersection.
– user14972
Commented Nov 30, 2016 at 22:10
• @Hurkyl - true. I was hurrying when I added that comment and went with the simplest notation. Between mine, yours, Arthur's, and Rohan/Limzy's, I think we have all the common notations covered now. Commented Dec 1, 2016 at 0:02

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

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.

• Hey Rohan, thank you and also the rest of the people who shared information on this thread. As this is the post which helped me the most in my understanding I will keep the discussion here although I don't want to take credit from the surrounding answers which surely helped my understanding as well. I will now attempt this problem... It seems that for the 1a problem the union (part 1) would be [0,[5,6)] and for the intersection (part 2) would just be [0]? Commented Nov 30, 2016 at 16:09