Is $\{\varnothing\}$ a subset of $\varnothing$, the empty set? Formed a table for a competitive exam:
$\begin{array}{cc}
\text{True} & \text{False} \\
\hline
\varnothing\subseteq\varnothing & \varnothing \in \varnothing\\
\varnothing\subseteq\{\varnothing\} & \,\\
\varnothing\in\{\varnothing\} & \{\varnothing\}\in\varnothing\\
\{\varnothing\}\subseteq\{\varnothing\} & \{\varnothing\}\in\{\varnothing\}
\end{array}$
I wish to know if $\{\varnothing\} \subseteq \varnothing$?
 A: Hint: The question is asking if all the elements in $\{\emptyset\}$ are in $\emptyset$. Since $\{\emptyset\}$ has 1 element, namely $\emptyset$, we want to check if that element is in $\emptyset$ (as in, the set on the RHS of the inclusion statement).
What are the elements of $\emptyset$?
A: {$\emptyset$} is not a sub-set of $\emptyset$ 
What's more: if A is a set it's never the case that {A} is a sub-set of A. 
The latter is true because if you assume {A} is a sub-set of A, this implies that A belongs to A. But that would mean that you need A to define A (because a set is defined by giving its elements). OK, I am talking only about well-founded set theories here. But these are usually the theories studied in schools/universities, so you're safe to remember that fact above. 
A: Definition: $(A\subseteq B\iff (x\in A\,\Rightarrow\, x\in B))$.   
Here: $(x\in \{\varnothing\}\,\Leftrightarrow\, x=\varnothing\,\not\Rightarrow\, x\in\varnothing)$,   
since it is impossible to be an element of $\varnothing$, so $\{\varnothing\}\not\subseteq\varnothing$.  
More generally, $\{A\}\not\subseteq A$ (equivalently $A\not\in A$) for any set $A$, which is implied by the Axiom of regularity (a part of the usual ZFC, regarded as the most common and standard axiomatic set theory).
