On the page of binary relation,Wikipedia explains about sets and classes:
In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set $A$, that contains all the objects of interest, and work with the restriction $=_A$ instead of $=$.
Similarly, the "subset of" relation $\subseteq$ needs to be restricted to have domain and codomain $\mathscr{P}(A)$ (the power set of a specific set $A$): the resulting set relation can be denoted by $\subseteq_A$.
Also, the "member of" relation needs to be restricted to have domain $A$ and codomain $\mathscr{P}(A)$ to obtain a binary relation $\in_A$ that is a set. Bertrand Russell has shown that assuming $\in$ to be defined over all sets leads to a contradiction in naive set theory.
I doubt about the the middle sentence, I think it should be redefined as:
Similarly, the "subset of" relation $\subseteq$ needs to be restricted to have domain $\mathscr{P}(A)$ and codomain $\mathcal{P}(B)$ (the power set of a specific set $A,B$): the resulting set relation can be denoted by $\subseteq$.
For example let's say the set $A$ is $R$-related to the set $B$ denoted by $ARB$ if $A\subseteq B$,then such a binary relation is a subset of the Cartesian product of the two given sets, e.g.
$$R\subseteq\mathscr{P}(A) \times \mathscr{P}(B)$$
Are my thoughts wrong?If so,then why?
To be strict, the situation may be even more general. Not only can we have different sets in domain and codomain, but they need not be powersets of other sets.
If $X,Y$ are any sets of sets (e.g., the infinite subsets of $\Bbb N$, the finite finite subsets of $\Bbb C$ with prime power cardinality, the compact intervals in $\Bbb R$, ...), we can investigate the subset relation between eleemtns of $X$ and $Y$. (Admittedly, in most cases we'd consider the case where $X=Y$, though). That is, the relation $R=\{\,\langle a,b\rangle\in X\times Y \mid a\subseteq b\,\}$. Thus we restrict $\subseteq$ not to elements of some powersets, but rather to elements of the (otherwise arbitrary) sets $X$ and $Y$. The $R$ above might be carefully denoted as $\subseteq_{X,Y}$. The whole point, however, is that this distinction can in practically all contexts be considered practically irrelevant.