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?


Definition of a symmetric relation $R$: a relation $R$ on a set $X$ is
  said symmetric if for all $x,y\in X$ such that $xRy$, then $yRx$.

This statement says that IF $x$ and $y$ are related, then $y$ and $x$ are related.

The false statement is $xRy$ and $yRx$ for all $x,y\in X$.

This statement makes the assumption that $xRy$ has a meaning whatever $x,y\in X$, which is false in general.
