Notation Question: What Does \bigcup^{N}_{1}x_{n} Mean?

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In summary, the conversation discusses the notation \bigcup^{N}_{1}x_{n}, which represents the union of a collection of sets. The speaker clarifies that this notation is not limited to a finite number of sets, as N can also be infinite or uncountable. The speaker also mentions that this notation was used without prior explanation, causing some confusion.
  • #1
Asphodel
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I have something like this:

[tex]\bigcup^{N}_{1}x_{n}[/tex]

What am I looking at? Is this [tex]x_{1}\cup x_{2}\cup x_{3}\cup ...\cup x_{N}[/tex]?
 
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  • #2
Yes, it's a union over some collection of sets.
 
  • #3
Not necessarily because N can be infinite say...

x1 U x2 U x3 U ...

But also, N can be uncountable, which means listing them in whatever way is pointless and incorrect.
 
  • #4
Thanks, Folland started using it without prior explanation (that I noticed).

It's the only interpretation that I could think of that made sense, but I didn't want to guess since it's already part of one theorem and one definition and if I guessed wrong then everything I did using those could be thrown off.
 

1. What is the meaning of the symbol \bigcup?

The symbol \bigcup represents the union of sets, which is a mathematical operation that combines all the elements from multiple sets into a single set.

2. What does the superscript N and subscript 1 represent in \bigcup^{N}_{1}x_{n}?

The superscript N and subscript 1 indicate the range of values that the variable n can take. In this case, n can take on values from 1 to N, inclusive.

3. What is the significance of the variable x in \bigcup^{N}_{1}x_{n}?

The variable x represents a set of values that are being combined using the union operation. In this notation, x could represent a single set or a collection of sets.

4. How is \bigcup^{N}_{1}x_{n} different from \bigcap^{N}_{1}x_{n}?

The symbol \bigcap represents the intersection of sets, which is a mathematical operation that finds the common elements between multiple sets. Therefore, \bigcup^{N}_{1}x_{n} and \bigcap^{N}_{1}x_{n} are opposite operations.

5. Can you give an example of using \bigcup^{N}_{1}x_{n} in a real-life scenario?

One example could be combining the different ingredients in a recipe to create a final dish. Each ingredient represents a set of items, and the union operation would combine all the ingredients to create the final dish.

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