Prove that if every X $\displaystyle \in$ S is transitive, then $\displaystyle \bigcup S$ is transitive.
Definition. A set A is said to be transitive set iff every member of a member of A is itself a member of A:
$\displaystyle x \in a \in A \Rightarrow x \in A$.
This condition can be stated as:
$\displaystyle a \in A \Rightarrow a \subseteq A$.
To show $\displaystyle \bigcup S$ is transitive,
$\displaystyle x \in \bigcup S \Rightarrow x \in X \in S \Rightarrow x \subseteq X \in S$ (X is transitive set by hypothesis) $\displaystyle \Rightarrow x \subseteq \bigcup S$.
Thus $\displaystyle \bigcup S$ is transitive.