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?
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 $X\notin X$.