Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\in$A}.

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

share|improve this question
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 1

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.

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.