TITLE: Is empty set element of every set if it is subset of every set?
QUESTION [12 upvotes]: 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?

REPLY [11 votes]: 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.