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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given a transitive set$$a,$$ I can prove that $$\bigcup a$$ is also transitive, but I don't quite like my method because I must first prove $$a \subseteq \mathcal{P}(\bigcup a)$$ to get at $$\bigcup a\subseteq \mathcal{P}(\bigcup a).$$ Could you please show me a smarter proof?

share|cite|improve this question
Also, are you asking about a subset of a transitive set, or about the union over a transitive set? – Asaf Karagila Jul 3 '11 at 15:33
Asaf, I am interested in proving $$\bigcup a$$ is transitive, then $$\bigcup \bigcup a$$, etc. – user11750 Jul 3 '11 at 16:57
Asaf, I can't find the transparent check mark by the vote count. Sorry, I feel inadequate in navigating this site. – user11750 Jul 3 '11 at 17:00
When you have an answer which you want to accept, just below the arrow to downvote it there is a transparent check mark. Click it and it will be accepted. – Asaf Karagila Jul 3 '11 at 17:05
As for the union of a union, if $\bigcup a$ is transitive, by the proof below $\bigcup\bigcup a$ is also transitive, and so on. Also, you might want to edit the question to incorporate your exact question, and not just "approximations" to it and complements in the comments. – Asaf Karagila Jul 3 '11 at 17:06
up vote 3 down vote accepted

Suppose that $a$ is transitive. That is $b\in c\in a\implies b\in a$.

Now take $d\in b\in\bigcup a$, then $b\in c\in a$, therefore $b\in a$, therefore $b\subseteq\bigcup a$ therefore $d\in\bigcup a$.

To add on the "A subset of a transitive set is transitive" is clearly false:
Take the transitive set $4=\{0,1,2,3\}$ (where $0=\varnothing$ and $n+1=n\cup\{n\}$) then $\{3\}$ is a subset of $4$ but not transitive itself.

share|cite|improve this answer

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.