Symmetric relation definition - why is this false? Can someone explain to me why the following statement is false, according to my study materials for discrete mathematics?

If a relation $R$ on a set $X$ is symmetric, then $x\,R\,y$ and $y\,R\,x$ for all $x,y\in X$.

How can this be false? I thought that if a relation is symmetric, then you need to have $xy$ and $yx$ in both directions, for all elements of a set.
This is the answer provided in the book:

False. The definition states that in order for $R$ to be symmetric on $X$, we have that if $x\,R\,y$ for $x, y \in X$ then $y\,R\,x$ also. This not imply that all $x$ and $y$ in $X$ are related under $R$.

What am I missing here?

The textbook is right.
For example: $A=\{1,2,3,4,5,6,7,8,9\}$ and $R_1=\{(1,1),(2,3),(3,2),(4,3),(3,4)\}$ and $R_2=\{(2,2),(3,3),(1,2),(4,1),(3,4)\}$
Here $R_1$ is symmetric since $\forall\ (x,y) \in R_1$ there exists $(y,x) \in R_1$ but $R_2$ is not since $\forall\ (x,y) \in R_2$ there does not exist $(y,x) \in R_2$.
The definition of symmetry does not involve the set $A$. It is as the book says and I have mentioned above.
