Is "empty set" an element of a set? Is "empty set" an element of a set? For example:
is this FALSE?:
0 ∈ {1,2,3}

 A: The empty set can be an element of a set, but will not necessarily always be an element of a set.
E.g.
$\emptyset \in \{\emptyset, \{a\},\{b\},\{a,b\}\}$
$\emptyset \in \{\emptyset, 1, 2\}$
$\emptyset \in \{A\}$ when $A=\emptyset$
There exist many sets though which the empty set is not a part of:
$\emptyset\not\in \{1,2,3\}$
$\emptyset\not\in \{x,y\}$
$\emptyset\not\in \emptyset$
$\emptyset\not\in\{\{\emptyset\}\}$
What will be true however is that the empty set is always a subset of (different than being an element of) any other set.
$\emptyset \subseteq \{1,2,3\}$
$\emptyset\subseteq \{a,b\}$

Additional details spawned from conversation in comments.
$\emptyset$ is the unique set with zero elements.  $\{\emptyset\}$ is a set with one element in it, the element namely being the emptyset.  Since $\{\emptyset\}$ has an element in it, it is not empty.  $\emptyset\neq \{\emptyset\}$
A set $A$ is a subset of another set $B$, written $A\subseteq B$, if and only if for every $a\in A$ you must also have $a\in B$.  In other words, there is nothing in the first set that is not also in the second set.
Here, we have $\{\emptyset\}\not\subseteq \{1,2,3\}$ since there is an element of the set on the left, namely $\emptyset$, which is not an element of the set on the right.
A: The answer is "It depends".  Some sets have the empty set as a member, other sets (like the example) you have given it isn't a member.
A: $\emptyset$ is a member of some sets, not of others. It is a member of $\{\emptyset\}$, for example. It is not a member of $\{1,2,3\}$ — which one of $1,2,3$ is $\emptyset$? (answer: none of them). It IS a subset of every set. 
