During class we had to come up with relations that are reflexive, but are not transitive or similar. I know the definitions of these terms. If we have a relation defined by the triple
$$r = (A, B, R)$$
where $R$ is a subset of the cartesian product
$$R \subseteq A \times B = \{ (a, b) \hspace{0.1cm} | \hspace{0.1cm} a \in A, b \in B \}$$
it is
$$\text{reflexive: if} \hspace{.1cm} x \hspace{.1cm} r \hspace{.1cm} x, \forall \hspace{.1cm} x \in A$$
$$\text{transitive: if} \hspace{.1cm} x \hspace{.1cm} r \hspace{.1cm} y \hspace{.1cm} \text{and} \hspace{.1cm} y \hspace{.1cm} r \hspace{.1cm} z, \text{then} \hspace{.1cm} x \hspace{.1cm} r \hspace{.1cm}z, \hspace{.1cm} \forall \hspace{.1cm} x, y, z \in A$$
$$\text{symmetric: if} \hspace{.1cm} x \hspace{.1cm} r \hspace{.1cm} y, \text{then} \hspace{.1cm} y \hspace{.1cm} r \hspace{.1cm} x, \hspace{.1cm} \forall \hspace{.1cm} x, y \in A$$
However, I couldn't come up with any such example. Are there any such examples on the basic numeric sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$?
$\{ (x,x), (y,y), (z,z), (x,y), (y,z) \}$