# Math Help - Transitivity

1. ## Transitivity

Prove that if every X $\in$ S is transitive, then $\bigcup S$ is transitive.

2. Originally Posted by qwe123
Prove that if every X $\in$ S is transitive, then $\bigcup S$ 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:
$x \in a \in A \Rightarrow x \in A$.

This condition can be stated as:
$a \in A \Rightarrow a \subseteq A$.

To show $\bigcup S$ is transitive,
$x \in \bigcup S \Rightarrow x \in X \in S \Rightarrow x \subseteq X \in S$ (X is transitive set by hypothesis) $\Rightarrow x \subseteq \bigcup S$.

Thus $\bigcup S$ is transitive.