What do $\cup_{A\in B}A$ and $\{T_A\}_{A\in B}$ mean?

Question

I'm fairly new to Set Theory and thus don't understand much of the notation. I was wondering what the following expressions meant:

1. $\cup_{A\in B}A$
2. $\{T_A\}_{A\in B}$

Answer 1

The first assumes that all the elements in $B$ are themselves sets (this is standard practice in axiomatic set theory, since everything is a set). Then the notation refers to the union of all the elements in $B$, i.e., $\bigcup_{A\in B}A=\{x\mid \exists A\in B\colon x\in A\}$. In axiomatic set theory this (or something closely related) is taken as one of the axioms.

The second one is slightly ambiguous. It may either refer to a set or to an indexed set. If it refers to a set then it is simply the set $S$ characterised by $x\in S\iff \exists A\in B\colon x=T_A$. If it refers to an indexed set then (often, but not necessarily) it is assumed that $B$ is a set and that we are actually interested in the actual assignment $A\mapsto T_A$. The difference between the two options is mainly that the later allows repeated elements, while the former does not.

Answer 2

The first one is the union of all elements of $B$. This notation is analogous to the summation notation. In Axiomatic Set Theory it is simply $\cup{B}$.

I don't know the second one.

Answer 3

$\bigcup_{A\in B}A$ is a notation for $\cup B$ which is the unique set with:$$x\in\cup B\iff x\in A\text{ for some }A\in B$$

It might look weird but actually mostly we see notations like $\bigcup_{\lambda\in\Lambda}A_{\lambda}$ (which on its turn is a custom notation for $\cup\{A_{\lambda}\mid \lambda\in\Lambda\}$)

The case above is a special case of that where $A_{\lambda}=\lambda$.

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$\{T_A\}_{A\in B}$ denotes a so-called family of sets $T_A$.

If $B$ is a set (so not a proper class) then it can be interpreted as a function defined on $B$ prescribed by $A\mapsto T_A$.