Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

I can't understand a sentence in a textbook: if $x$ is a transitive set, then $\bigcup x^+=x$? Could someone help me to understand?

added: $x^+=x\cup\{x\}$

share|cite|improve this question
It helps if you tell us what the textbook is. – Paul Jan 11 '12 at 8:13
It is a chinese book. Maybe it is wrong. – Paul Jan 11 '12 at 8:16
up vote 5 down vote accepted

Let us deconstruct this:

  1. $\bigcup A = \{z\mid\exists y\in A: z\in y\}$.
  2. $x^+=x\cup\{x\}$

From this we have: $\bigcup x^+ = \{z\mid\exists y\in x\cup\{x\}: z\in y\}$. We then follow the definition of a transitive set:

The set $x$ is transitive if for every $y\in x$, $y\subseteq x$

Now we want to show that $x\subseteq\bigcup x^+$ and that $\bigcup x^+\subseteq x$, the axiom of extensionality will ensure equality.

For the first one, it is nearly trivial. Since $x\in x^+$ we have that $y\in x$ then $y\in\bigcup x^+$ immediately (recall the definition of this union).

For the second one, we recall that $y\in x^+$ then $y\in x$ or $y=x$. Therefore $y\subseteq x$. So for every $z\in\bigcup x^+$ we have some $y\subseteq x$ such that $z\in y$, therefore $z\in x$ as wanted.

share|cite|improve this answer

The set $x$ is transitive if $z\in y \in x$ implies $z\in x$. Moreover, $x^+=x\cup\{x\}$. So let $x$ be a transitive set.

We have $x\subseteq\bigcup x^+$ since $x\in\{x\}$. Now let $z\in \bigcup x^+$. Then $z\in\bigcup x\cup x$. If $z\in\bigcup x$, then there exists $y\in x$ with $z\in y$. But since $x$ is transitive, $y\subseteq x$ and hence $z\in x$.

share|cite|improve this answer
There was a typo I corrected, please take a look at the edited proof. – Michael Greinecker Jan 11 '12 at 9:06

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.