Can a relation from A to some other set B also be considered symmetric?

Question

Note: This definition is from Discrete Mathematics and Its Applications [7th ed, page 577].

This is my book's definition of a relation R on a set A

My question is does the definition of a symmetric relation extend to a relation from a set A to another set B, that doesn't have to be A?
Say you have two sets, A {1,2} and set B {1,2,3}
Suppose a relation R from the set A to the set B is the empty set, {} (legit relation because the empty set is a subset of A x B)
From what I learned here Symmetric, the empty relation is technically symmetric because there are no elements (a,b) to violate the symmetric definition. Now would this relation symmetric even if it isn't a relation from A to A?

Answer 1

Usually the word "symmetric" is only used about relations from a set to itself. If the relation is from a set to a different set, we don't call it symmetric -- not because it fails to satisfy the condition, but because the question "is it symmetric or not?" is not supposed to be asked about relations between different sets in the first place.

However, if you want to use the word about relations from $A$ to $B$ that happen to satisfy the formal condition, nobody's stopping you. But the burden will be on you to make sure that you're not being misunderstood when you do so (some readers may think you're implicitly saying that the relation is from $A$ to $A$ when you call it "symmetric"), and also to avoid using results that are formulated as "Let $R$ be a symmetric relation. Then such-and-such." which actually hold only when $R$ is a relation from $A$ to $A$, as is commonly implied.

Answer 2

Suppose we had such a symmetric relation between A and B. Then, for each element $a\in A$ so that $a\,R\,b$ for $b\in B$, it follows that $b\,R\,a$. Hence, $b\in A$ and $a\in B$. This means that $a,b\in A\cap B$ for $a,b$. Thus, this relation could really just be considered a symmetric relation on $A\cap B$. So even if you have defined such a symmetric relation between two sets, it's really just a symmetric relation on a single set.