I would guess the relation R would be symmetric if A was equal to {b,c,d}, but not {b,c,d,e} as in the textbook?

But let me try to reason through it.

Let's say we consider eRb. Then if the relation is symmetric, it must follow that bRe. But since there's no (e,b) in R (i.e. ~eRb), the statement $ \forall x,y \in A, xRy \implies yRx $ is vacuously true for x=e, y=b, and so with any other pair involving e. Hence R is symmetric. Is that correct?