For relations to be reflexive, symmetric and transitive is the property true for just the single subset $A$ or $A\times A$? I was going over my notes on what it means for relations to be reflexive, symmetric and transitive and I'm unclear on one thing: is it for every $x$ in a set $A$ or set $A\times A$? So my understanding of the definitions are
A relation $R$ on a set $A$ is


*

*reflexive if $(x,x) \in R$ for every $x \in A$ 

*symmetric if $(y,x) \in
   R$ whenever $(x,y) \in R$ for every $(x,y) \in  A \times A$

*transitive if $(x,z) \in R$ whenever $(x,y) \in R ,(y,z) \in R$  for every
$x,y,z \in A$


I'm unclear why is it sometimes "$... \in A$" and other times "$... \in  A \times A$"? 
Are my notes wrong?
 A: A relation is always a subset of a cartesian product. In the present case, $\;R\subset A\times A\;$, and then we talk of "a relation on the set $\;A\;$", but the actual meaning is the above mentioned.
Thus, if we've a relation as above, we say it is reflexive if $\;(x,x)\in R\;\;\forall\,x\in A\;$ , as we understand both entries in each ordered pair are taken from the same set $\;A\;$, and likewise for symmetric, transitive, etc.
A: "$(x,y) \in A \times A$" means exactly the same thing as "$x,y \in A$".  In my opinion, the definition would have been less confusing if whoever wrote the definition had written  "$x,y \in A$".  Also "$x \in A$" is equivalent to "$(x,x) \in A \times A$", but it clearly would be silly to write it that way, and fortunately whoever wrote the definition to not choose to do that.  Finally, "$(x,y,z) \in A \times A \times A$" is equivalent to "$x,y,z \in A$", but the latter is less confusing.  
BTW: Brian M. Scott explains this very well in his comments.  Your question is really a notational question, and has nothing to do with properties of relations.
