What is a transitive set? Show that if $x$ is transitive then so are the union $\bigcup x$ and the power set $P x$ of $x$. If $\bigcup x$ is transitive, is $x$ transitive? If $P x$ is transitive, is $x$ transitive? Justify your answers.

What is the transitive closure of a set? Show that any set $x$ has a transitive closure $T C(x)$.

Suppose that $x$ has rank $\alpha$. What is the rank of $P x$ ? What is the rank of $T C(x)$ ?

[You may use standard properties of rank.]