In set theory, the notation $$\bigcup X$$ means the union of all elements of $X$. For example, $\bigcup\{ a,b \}=a \cup b$.

I encounter the following notation $$\bigcup X \subseteq X$$ in the book 'Introduction to Set Theory' by Hrbacek and Jeck. Does it make sense?

For me, it doesn't make sense. Instead, it should be $$\bigcup X \in X$$

Am I doing anything wrong?

  • $\begingroup$ Suppose $$ X=\{\{a\},\{b\}\}. $$ Then $$ \bigcup X=\{a\}\bigcup\{b\}=\{a,b\}. $$ So as you can see, $\cup X\not\in X$. $\endgroup$ – Kim Jong Un Sep 22 '14 at 16:46
  • $\begingroup$ I suspect that the book meant in this particular case. For example, if $X$ is an ordinal, then $\bigcup X \subseteq X$. $\endgroup$ – Thomas Andrews Sep 22 '14 at 16:52

It is possible that $\bigcup X\subseteq X$, and it is possible that not. It is also possible that $\bigcup X\in X$ and it is also possible that not.

If $\bigcup X\subseteq X$, we call $X$ a "transitive set". $\varnothing$ is a transitive set, so is $\{\varnothing\}$. In fact, if $X$ is an ordinal then $\bigcup X\subseteq X$.

You might be tempted to say in the case that $X$ is an ordinal that $\bigcup X\in X$ is true, but that is only true when $X$ is a successor ordinal, then $\bigcup X=\max X$ (remember that an ordinal is itself a set of ordinals). If $X=\omega$, for example, then $\bigcup X=X$, in which case $X\notin X$.


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