Arbitrary union example

1. Yes, that's correct.

2. It's $x\cup\{x, y\} $, the set of all elements inside $x$ and the elements $x$ and $y$ themselves.

If $A$ is a set with two elements, then $\bigcup A$ is the union of these two elements.

1. $\{\{x\},\{x,y\}\}$ has two elements, so $\bigcup\{\{x\},\{x,y\}\}=\{x\}\cup\{x,y\}=\{x,y\}$.
2. $\{x,y\}$ has two elements, so $\bigcup\{x,y\}$ is indeed $x\cup y$.
3. How many elements does $\{x,\{x,y\}\}$ have? Well, two: $x$ and $\{x,y\}$. Therefore $\bigcup\{x,\{x,y\}\}$ equals...