If an empty set is an element of a set, $\{5,\{\}\}$ is that equal to just $\{5\}$? Is this true $\{5, \emptyset\} = \{5\}$?
I know that the empty set is always a subset of any set, but when it's an element is that necessary to write in or not?
 A: No, you are confusing being an element with being a subset.  It is true that $\emptyset \subset \{5\}$ but it is not true that $\emptyset \in \{5\}$.  This is an important distinction.
A: No. The empty set is still a set, not nothing.
A set is always something, in this case it's just a set containing nothing.  
It's a subset of all sets, because you can always take out all their elements, but not an element since the set doesn't have an element that's a set with no items.
Therefore, no, they are not equal.
A: It is not true.
$B \subset A$ means that $B$ is a subset of $A$, i.e. every element in $B$ is also in $A$. $e \in B \Rightarrow e \in A$.
So because the empty set $\emptyset$ contains no objects, $\emptyset \subset A$ for any $A$.
$\emptyset \in A$ is true only when $A$ contains the element "empty set", i.e. when $\{\emptyset\} \subset A$.
Let $B, C$ be sets. $A = \{B, C\}$ is the set containing as elements the two sets $B$ and $C$. $B$ and $C$ are elements of $A$.
$A' = B \cup C$ is the union of $B$ and $C$, i.e. the set containing all elements of $B$ and of $C$. $B$ and $C$ are subsets of $A'$.
