Notation question in elementary set theory - what is $\bigcup A$?

Question

Let $A$ be a set. What is defined as $\bigcup A$? Is it the union of all sets that $A$ includes? Could someone provide an example for this notation?

Thank you!

Answer 1

$$x\in \bigcup A\iff x\in a\text{ for some }a\in A$$

Actually: $$\bigcup A:=\left\{ x\mid\exists a\in A\; x\in a\right\} $$

Quite often you meet notations like: $$\bigcup_{i\in I}A_{i}$$ This could also be written as: $$\bigcup\left\{ A_{i}\mid i\in I\right\} $$

A set $A$ is called transitive if $x\in a\wedge a\in A$ implies that $x\in A$. This is exactly the statement that $\bigcup A\subset A$.