For relations to be reflexive, symmetric and transitive is the property true for just the single subset $A$ or $A\times A$?

Question

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?

Answer 1

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.

Answer 2

"$(x,y) \in A \times A$" 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.