Small notation question about union of chains (Set Theory)

Question

The question is derived from this question I encountered:

Let $A$ be a set, and let there be a function $f: A \rightarrow A$, so that for every $a \in A$, $f(a) \neq a$.

Define $S=\{X \subseteq A: X \cap f(X)=\emptyset \}$. Show that if a chain $C=\{X_a: a \in I \}\subseteq S$, then $\cup C \in S$.

However, the meaning of $\cup C$ is unclear to me. $C$ is a set of sets, and $\cup C$ is a set made of the union of these sets? So it's a single set?

Thanks for any assistance!

Answer 1

Note that, by definition, $x \in \bigcup C$ iff there is some $y \in C$ such that $x \in y$. In your specific example, $\bigcup C$ consists precisely of those elements contained in some $X_a$, where $a \in I$.

So, if for example $C = \{X_1,X_2,X_3\}$, where $X_1 = \{1\}$, $X_2 = \{1,2\}$, $X_3 = \{3,4\}$, then $\bigcup C = \{1,2,3,4\}$.