Is empty set element of every set if it is subset of every set? This problem is from Discrete Mathematics and its Applications

My question is on 9b. I know that the sign represents an element is a member of.
(from book)

I know that the O with a slash across it is the empty set which "is a special set that has no elements".
From http://mathcentral.uregina.ca/QQ/database/QQ.09.06/narayana1.html, I got that the empty set is a subset of all sets, meaning that every member of the empty set(nothing) is also a member of any other set.
Based on all of this, for 9b, would {0} contain the empty set because it fundamentally has the elements that consist of the empty set(nothing) or does it physically have to
have the empty set?
 A: $x \in \{ y \} $ if and only if $x = y$. Thus, $\varnothing \in \{ 0 \}$ if and only if $\varnothing = 0$.
Out of context, there is actually ambiguity here. Often, in set-theoretic contexts, we interpret natural numbers as being the set of all smaller natural numbers; e.g. $3 = \{ 0, 1, 2 \}$. And according to this convention, $0$ is indeed equal to $\varnothing$.
But we might not adopt this convention, and we take $0$ to be its own thing that is unequal to $\varnothing$ or any other set that is 'naturally' written.
A: When $X$ and $Y$ are two sets, we say that $X\subset Y$ if every element of $X$ is contained in $Y$.
With this definition, you see that $\emptyset \subset Y$ for any set $Y$. Indeed, there is no element in $\emptyset$, so every element of $\emptyset$ is contained in $Y$ (trivially true as there is nothing to check).
However, if you want to write $\emptyset \in Y$, this means that there is one element of $Y$ which is a set and that this set is the empty set. When $Y=\{0\}$, you have only one element in $Y$, and this one is not a set, it is a number, which is $0$. Hence, $\emptyset\notin \{0\}$.
Both statements $9a$ and $9b$ are false.
A: There are several ways to represent the empty set. $\{ \}, \emptyset, \text{ and } \varnothing$ are three common ways.
Saying "the empty set(nothing)" is incorrect. The empty set is the set that contains nothing. A bottle can contain nothing, but the bottle itself is something.
Hence, for example, the set $\{\varnothing\}$ is not the empty set simply because it has something in it. In English, the set containing the empty set is not the empty set.
For the empty set to be a member of a set, it has to actually be in that set. The empty set is in $\{1,2,\varnothing\}$. The sets $\{1,2\}$ and $\{1,2,\{\varnothing\}\}$ do not have the empty set in them.
A subset of the set S, is either the set S or the set S with some stuff removed from it.
For example, a subset of $\{a,b\}$ is the set $\{a,b\}$ with $0$ to $2$ things removed from it. These sets are subsets of $\{a,b\}$:
\begin{align}
  \{a,b\} &- \text{nothing was removed}\\
  \{b\}   &- \text{a was removed}\\
  \{a\}   &- \text{b was removed}\\
  \{ \}   &- \text{a and b were removed}
\end{align}
where the last set, $\{ \}$, is the empty set.
Start with any set, take everything out if it, and you are left with an empty set. Hence the empty set is a subset of every set.
A: No.  Elements and subsets are not the same thing.
An element of a set is one of the things in the set.
A subset of a set is another set that does not contain any elements which are not elements of the set to which it is a subset.
The empty set is not an element of {1,2,3}.
That is because 1 is not the empty set, 2 is not the empty set, and 3 is not the empty set.
"Empty set" does not mean "nothing".  Instead, it is the set that does not have elements.
A: Recall the (naive) definitions of the symbols $\subset$ and $\in$. The reason that $\varnothing\subset A$ for any set $A$ is because any "$x$ in $\varnothing$" is automatically also in $A$ (vacuously, because there is no such $x$). 
On the other hand, $\varnothing$ is an element of, say, the set $\{ \varnothing\}$, for the same reason that $1\in \{1\}$. 
A: Lets consider this situation:
     Let set $A = {1,2,3}$, if null/empty set is an element of any set, therefore the cardinality of set A is 4 which is not true from the definition of cardinality of set which states that it is the number of elements in a given set. The truth is, set A has a cardinality of 3 since it has only three elements and it has 8 subsets. In other words, null set is not an element of any set but a subset of any set.
