Relations from X to Y

Question

Let $X$ and $Y$ be sets, and let $R$ be a binary relation from $X$ to $Y$.

What does it mean for $R$ to be reflexive and symmetric? Because I know that for a relation over a single set $A$:

It is reflexive if:

For all $x\in\mathbb A, xRx$

And it is symmetric if:

For all $x,y\in\mathbb A$, if $xRy$ then $yRx$.

What I can't understand or find any information on is how these definitions carry over for relations over sets $X$ and $Y$ that are not equal.

Answer 1

Assume R is symmetric.
If x in X, y in Y, xRy, then yRx, y in X, x in Y.
So domain R subset range R; range R subset domain R;
domain R = range R.
Thus R subset (X $\cap$ Y)×(X $\cap$ Y).

Consequently for symmetric R, R is reflective when
for all x in X $\cap$ Y, xRx.