Is $\{a, b\}$ a subset of $\big\{\{a, b\}\big\}\;?$

Question

I read a lot of questions like that here, but it takes me more confused.

In many books appear examples like:
$\textbf{A} = \{a, b, c\}$, so: $\{a\}\subset\textbf{A}$ , $\{b, c\}\subset\textbf{A}$ etc.

But here I read things like:
If $\;\textbf{B}=\big\{a,b,\{c, d\}\big\}$, so $\textbf{B}$ has $3$ members:
$a\in\textbf{B}$ , $b\in\textbf{B}$ , $\{c, d\}\in\textbf{B}$ , but :
$c\not\in\textbf{B}$ and $d\not\in\textbf{B}$.

It made me more confused because:
If there is $x\in\textbf{P}$ and $\textbf{P}\subset\textbf{Q}$, so $x\in\textbf{Q}$.
E.g.: $1\in\mathbb Z$ and $\mathbb Z\subset\mathbb R$, so $1\in\mathbb R$.

In this case:
$\{c, d\}\subset\textbf{B}$ ?
$\big\{\{c, d\}\big\}\subset\textbf{B}$ ?
$\{c\}\subset\textbf{B}$ ?
$\big\{\{c\}\big\}\subset\textbf{B}$ ?

Answer 1

The Definition:

$A \subset B \hspace{10pt}$ iff $\hspace{10pt} \forall x \hspace{3pt}\big(\hspace{2pt}x \in A \implies x \in B \hspace{2pt}\big)$

That is,
$A\subset B\hspace{5pt}$ iff $\;$ every element of $A$ is also an element of $B$.

Let's take example in the title: $A = \{a, b\}, B = \big\{\{a, b\}\big\}$

$A$ has two elements: $a$ and $b$.
$B$ has only one element: the set $\{a, b\}$

Is $a $ in $B$? No, it is not! $B$ has only one element that is the set $\{a, b\}$ which is not the element $a$.
Hence, it is clear that an element of $A$ is not an element of $B$.
Thus $A$ is not a subset of $B$.