Meaning of \bigcup Symbol & Index Set Explained

[Yegor]

T is an index set. And for each $$t \in T$$ $$A_t$$ is a set
$$\bigcup_{t \in T} A_t = \{x : \exists t \in T with x \in A_t \}$$
What means this $$\bigcup$$ symbol and entire expression?
And question on index set: is it used just for orderring any other set?

 

[Galileo]

It's the union. If A and B are sets, $A\cup B$ denotes the union of A and B. It's that set which contains all the elements of A and those of B. So
$$A \cup B = \{x|x\in A \vee x\in B\}$$

To generalize this to a union of an arbitrary number of sets is easy. That's exactly what your expression is: the union of all $A_t$.

 

[Yegor]

thank you very much, Galileo!
What about my guess about "index set"?

 

[Galileo]

I didn't understand what you meant exactly, but I think you have the right idea. The index set is just there to label the other sets. This way you can make T finite, countably infinite or uncountably infinite with the same notation. So the collection of sets A_t may be a finite, or infinite collection of any cardinality.

 

[dmehling]

I just stumbled onto this post and it relates exactly to what I'm trying to figure out. This concept of an index set is very baffling to me. Can you give a little more detail on what exactly an index set is?