$\{(1,2),(2,3),(1,3),(3,1)\}$ 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!
Best Answer
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).