Some questions in set theory about belonging Some questions about set theory:
1.If $a$ is an element of a set $A$, is it a subset of $A$ or not?
2.If not, can $A$ be a subset of $a$ ?
 A: 1) It may or may not be. For instance, $A=\{\emptyset\}$ is a set with a single element $a=\emptyset$. In this case it is also the case that $a\subseteq A$, namely $\emptyset \subseteq A$, simply because $\emptyset \subseteq B$ for any set $B$. However, $A=\{1\}$ is also a set with a single element, namely $a=1$. In this case $a\nsubseteq A$, simply becuase $a$ is not even a set so one can't speak of it being a subset (emphasis here on set) of anything. 
2) If $a\in A$, then if $A\subseteq a$ then any element of $A$ in an element of $a$. In particular, $a$ must be a set and, since $a\in A$ it follows that $a\in a$. We now get into the question of which axioms you choose for set theory. Many axiomatizations disallow $a\in a$, but some, called non-well-founded sets, may allow it. So, the answer is, naively this is not allowed, but the notion of such sets is studied. 
A: Example: $$A=\{3,\{4\},5,\{5\}\}$$  $3$ is an element, but not a subset, of $A$. $\{3\}$ is a subset, but not an element, of $A$.  $4$ is neither an element nor a subset of $A$, but $\{4\}$ is an element, but not a subset, of $A$.  $5$ is an element, but not a subset, of $A$.  $\{5\}$ is both an element and a subset of $A$.
For the second question, the answer is "maybe".  The axiom of regularity, which is part of the widely used Zermelo-Frankel axioms, prohibits $A$ to be an element of itself, which I believe excludes this possibility.  The short answer is that no familiar sets will admit this property.
