Understanding notation for binary relations Determine whether or not following relation is reflexive, symmetric, anti-symmetric and transitive. $A$ is an arbitrary set:
$$R=\left\{(x,y)\in{\bigl(\mathscr P(A)\bigr)}^2\mid x\cap y\neq\emptyset\right\}.$$
I am having trouble understanding the notation here. The condition to check reflexivity is to check that for all elements $x$ in $A$, $(x,x)$ is in $R$. But I am confused about how an element in the Cartesian product of the powerset of $A$ could have the form $(x,y)$, where $x$ and $y$ are elements of $A$.
Basically I am looking for an English translation of the statement 
$R=\left\{(x,y)\in{\bigl(\mathscr P(A)\bigr)}^2\mid x\cap y\neq\emptyset\right\}$. What exactly is this statement saying about the properties of $R$?
 A: The relation $R$ says that $x \sim y$ (i.e. $x$ is related to $y$), where $(x,y) \in P(A) \times P(A)$, if $x \cap y \neq \emptyset$.
Here, by definition, $x$ and $y$ are elements of $(P(A))^2$ (i.e. $P(A) \times P(A)$).
In order to check reflexivity, for example, you would check that $x \sim x$, for $x \in P(A)$. 
I.e. check that $(x,x) \in R$. This is clear, though, because $(x,x) \in (P(A))^2$, and $x \cap x \neq \emptyset$. By definition of $R$, we have indeed that $(x,x) \in R$.
A: $R= \{ (x,y) \in P^2(A) : x \cap y \ne  \emptyset \}$
$(x,y) \in P^2(A)$ means that $x$ and $y$ are subsets of $A$. Hence $xRy$ if and only if $x$ and $y$ have at least one point in common.
At the urging of @HenningMarkholm, 
$\mathcal{P}^2(A) = (\mathcal{P}(A))^2 =\mathcal{P}(A) \times \mathcal{P}(A) $ where $\mathcal{P}(A)$, the power set of $A$, is the set of all subsets of $A$. So $(x,y) \in \mathcal{P}^2(A)$ means that $x \in \mathcal{P}(A)$ and 
$y \in \mathcal{P}(A)$. In other words, $x$ and $y$ are subsets of $A$.
