# Is relation $\{(1,2),(2,3),(1,3),(3,1)\}$ symmetric and transitive?

$\{(1,2),(2,3),(1,3),(3,1)\}$ is our set.

According to this set, I know that it isn't reflexive. Because;

$\{(1,1),(2,2),(3,3),(4,4)\}$ are missing.

However, I also think that it's symmetric and transitive. Transitive because;

$\{(1,2),(2,3),(1,3)\}$

Symmetric because;

$\{(1,3),(3,1)\}$

Is it correct or to be symmetric, should each one of the subsets be symmetric? Like;

$\{(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)\}$

Thanks!

• By definition, in order to say that a relation $R$ is symmetric, you should for all $x$ and $y$ have if $xRy$ then $yRx$. You have $(1,2)$ in the set, thus $(2,1)$ should also be contained which is not the case. Thus the relation is not symmetric. It is also not transitive since $(2,3)$ and $(3,1)$ are in the set but $(2,1)$ isn't. – Levent Dec 18 '16 at 14:07

It is not symmetric because for example $(1,2)$ is in the set but $(2,1)$ is not.

It is not transitive also because $(1,3)$ and $(3,1)$ are in the set, but $(1,1)$ is not.

The $\{(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)\}$ will be indeed symmetric, but still not transitive. You still need to add $(1,1)$, $(2,2)$ and $(3,3)$ to make it transitive, and in this case it will also become reflexive on $\{1,2,3\}$, however it will not be reflexive on $\{1,2,3,4\}$ (for that you would need to include $(4,4)$ as well).