Unions and intersections of indexed families of transitive sets are transitive

Let $\{ T_a \}_{a \in A}$ be a family of transitive sets. Prove that $\bigcup_{a \in A} T_a$ and $\bigcap_{a \in A} T_a$ are transitive. Assume $A \neq \emptyset$.

I'm not sure how to apply the definitions to write the proof.

For the union we will have $\bigcup$ $T_a$= {x|x$\in$$T_a for some a \inA}. For the intersection we will have \bigcap T_a= {x|x\in$$T_a$ for every a $\in$A}.

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What did you try and where did you get stuck? You just need to verify the definitions, it's quite easy, actually. – Asaf Karagila Jan 27 '14 at 21:31

1 Answer

Write $T=\bigcup T_a$, for simplicity. You want to show that $T$ is transitive, then you need to show that a certain definition applies for $T$. That is, if $x\in T$ and $y\in x$, then $y\in T$.

Suppose that $x\in T$, then there is some $a\in A$ such that $x\in T_a$. What do you know about $T_a$? Conclude that $y\in T$.

Similarly, apply the definitions for the intersection.

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I'm confused how you're applying the definition of union to the indexed family... can you explain this please? – user2553807 Jan 27 '14 at 22:27
What's the confusion exactly? – Asaf Karagila Jan 27 '14 at 22:30
I don't see where the union and intersection come into play in what you've written... all I can identify is the transitive property. – user2553807 Jan 27 '14 at 22:46
Yes, $x\in T\iff x\in\bigcup T_a\iff\exists a\in A: x\in T_a$. – Asaf Karagila Jan 27 '14 at 22:47