In “An element of a set can never be a subset of itself”, what does ‘itself’ stand for? I have just begun learning about sets. My first language isn't English. I'm in high school.
Here's an example problem I found in my textbook:

Example 11: Let $A, B$ and $C$ be three sets. If $A∈B$ and
$B⊂C$, is it true that $A⊂C$? If not, give an example.
Solution: No. Let $A=\{1\}, B=\{\{1\}, 2\}$ and $C=\{\{1\}, 2, 3\}$. Here $A∈B$ as $A=\{1\}$ and $B⊂C$. But $A⊄C$ as $1∈A$ and $1∉C$.
Note that an element of a set can never be a subset of itself.

The link to the textbook's chapter. 
What does “itself” stand for here? Does it mean an element of a set can't be it's own (the element's) subset?
Or does that mean an element cannot be both an element and a subset of a set at the same time?
If $P=\{p\}, Q=\{\{p\}, q\}$, and $R=\{\{p\}, q, r\}$, we can say that $P∈Q$. But, can we say that both $Q∈R$ and $Q⊂R$ are true? Is it so that $Q$ cannot be both an element and a subset of $R$? Is $\{\{p\}, q, r\}$ not the same as $\{p, q, r\}$?
 A: It helps to think of the braces $\{\}$ as being quite literal. So if $P=\{p\}$, $Q=\{\{p\},q\}$, and $R=\{\{p\},q,r\}$, then:


*

*When we write $Q=\{\{p\},q\}$, it means that the set $Q$ contains the two elements $\{p\}$ and $q$. In symbols, $\{p\}\in Q$ and $q\in Q$. Since $P=\{p\}$, we can interchange those two things, so we can also write $P\in Q$.

*The statement "$P\subset Q$" means "any element of $P$ is an element of $Q$." Well, $p$ is an element of $P$, but not of $Q$.

*The set $R$ contains $\{p\}$, $q$, and $r$, and $Q$ contains $\{p\}$ and $q$. Thus, $Q\subset R$. However, $Q\notin R$, because $R$ does not contain the element $Q=\{\{p\},q\}$.


As someone pointed out, it is not true that if $x\in S$, then $x$ is not a subset of $S$, nor the similar statement that if $x\subset S$, then $x\notin S$. The set $\{1,\{1\}\}$ gives a counterexample to both. The only true statement I can think of here is that a set $S$ is never an element in itself. We can never have $S\in S$.
A: Both interpretations are sensible. Unfortunately, both interpretations are false statements! That comment is just misguided. (It's not your fault; it's the author's fault)
For instance: your first interpretation is:

If $A$ is a set, and $x\in A$ is an element of $A$, then $x$ cannot be a subset of $x$.

But that is false. In Set Theory, sets can be elements of other sets, and every set is a subset of itself. So $x$ can certainly be a subset of itself. For example, if $A=\{\{1\},\{2\}\}$, then $x=\{1\}$ is an element of $A$, and $x$ is a subset of itself.
Your second interpretation is:

If $A$ is a set, and $x\in A$, then $x$ cannot be a subset of $A$.

But that is also false. In fact, there is a whole class of sets, known as "transitive set", with the property that every element is also a subset. For instance, the set $A=\{\varnothing,\{\varnothing\}\}$, whose elements are (i) the empty set and (ii) the set whose only element is the empty set; has the property that each of its elements is, in addition to being an element of $A$, also a subset of $A$. 
In short: I'm not sure what the author meant to say with that comment, but both natural interpretations of it are false.

What is true is that, in general, if $A$ is a set and $x\in A$ is an element of $A$, then you cannot say, from these facts alone, whether $x$ is a subset of $A$ or not; and if your set theory allows for objects that are not sets ("ur-elements"), then you may not know whether $x$ is a subset of itself or not.
It is also true that in many set theories, one cannot have a set be an element of itself: that is, you can never have $A\in A$. (But there are set theories where this is valid, however...)
A: "Itself" means the "set" here.
Example- A={a,b,c}
Therefore, according to the statement -"An element of a set can never be a subset of itself/the set.", a ⊄ A, b ⊄ A and c ⊄ A, which is true because a, b, c are the elements of A and not the subsets of A.
However, {a} ⊂ A, {b} ⊂ A, {c} ⊂ A, where the left hand side of "⊂" are some sets which has the element which is also present in A.
(Remember: A subset is that set whose elements are also present in another set(superset), so, because the elements are themselves not sets, so, they can't be subsets but the set of those element(s) can be a subset.)
