Are these an/a $∈$ or $⊆$ of the following set? Let $A = \{4,5,6,7\}$. Write $∈$ or $⊆$:
$\{4\}\underline{\qquad}A$  
$\{5,6\}\underline{\qquad}P(A)$  
${∅}\underline{\qquad}P(A)$  
$5\underline{\qquad}A$  
$A\underline{\qquad}P(A)$  
${A,∅}\underline{\qquad}P(A)$
I know the difference between $∈$ and $⊆$. I simply do not know what difference does it make if a number is surrounded by the curly brackets or not? (Such as the case with $\{4\}$ and $5$). Further example, I know that that the powerset $P(A)$ contains ${A}$, but what is the relation between $A$ and $P(A)$? Is $A$ and $\{A\}$ any different? And is ${A,∅}$ the same as $\{A\}$?
 A: We say that $x \in A$ if $x$ is an element of a set $A$.
If $A \subseteq B$ then $x \in A$ implies that $x \in B$ (If $A$ is a subset of $B$ then $A$ contains the same elements as $B$).
Example: Let $A = \{1\}$ and $B = \{ 1, 2\}$, then $A \subseteq B$ since $1 \in A$ and $1 \in B$.
It is the case that $A \ne \{A\} $ $(A = \{4,5,6,7\} \ne \{\{4,5,6,7\}\} = \{A\})$ since $A \in \{A\}$ and $A \not\in A$.  Also, $A \subseteq A$, and $A \not\subseteq \{A\}$. The set $\mathcal{P}(A)$ denotes the set of all subsets of $A$ and if $A$ conains $n$ elements, $\mathcal{P}(A)$ contains $2^n$ elements. 
$\{4\} \subseteq A$
$\{\emptyset \} \not\in \mathcal{P}(A)$ $($but $\begin{align} \emptyset \in \mathcal{P}(A))\end{align}$
$\{5,6\} \in \mathcal{P}(A)$
$5 \in A$
$A \in \mathcal{P}(A)$ (Since $A \subseteq A$, $A \in \mathcal{P}(A)$)
A: Using the fact that:
$$
X \in P(A) \iff X \subseteq A \iff \forall x \in X,\, x \in A
$$
and:
$$
\{x\} \neq x
$$
we find that:


*

*$\{4\} \subseteq A$ (since $4 \in A$), but $\{4\} \notin A$

*$\{\emptyset\} \subseteq P(A)$ (since $\emptyset \in P(A) \iff \emptyset \subseteq A$), but $\{\emptyset\} \notin P(A)$ (since $\{\emptyset\} \not\subseteq A \iff \emptyset \notin A$)


See if you can do the others.
A: If a number is surrounded by "curly brackets" it is a set. Yes, $A$ and $\{A\}$ are quite different. The first is just a single element $A$, whereas the latter is a set only containing the element $A$. Can you now see when you should use $\in$, and when you should use $\subseteq$?
I will answer the first one as an example. Let $A = \{4,5,6,7\}$, then $A$ does not contain any sets, so we can not say that $\{4\}\in A$, however $\{4\}$ is a subset of $A$, since all elements of $\{4\}$ are also contained in $A$, so $\{4\}\subseteq A$.
A: $A$ is en element of $\mathcal P(A)$; $A$ is different from $\{A\}$. The elements of $A$ is 4,5,6,7, however the element of $\{A\}$ is only $A$. $\{A, \emptyset\}$ is also different from $\{A\}$. One has two elements and the other has only one.
