First of all, I'm sorry for the terribly drawn image.
So I need to state whether the relations A, B, C, and D are reflexive, symmetric, and/or transitive. BUT, the four of them can't have the same properties.
So for A, I know that it is symmetric, but not reflexive and not transitive.
For B, it is reflexive, but not symmteric or transitive.
For C, it is symmetric, but not reflexive or transitive (this is the same as A?!!)
And finally for D, it's not reflexive and symmetric, but I'm not sure if it's transitive?
Would really appreciate some help. Thank you!
No, $D$ is NOT transitive.
You're correct that $D$ is not reflexive nor is it symmetric.
In fact, $D = \{(1, 2), (2, 3), (3, 1)\}.$
While it is true that $1\sim 2$ and that $2\sim 3,\;$ $1 \large{\not\sim} 3\;$ (i.e., $(1,3)\notin D$.) And therefore $D$ is not transitive.
The one other correction to make is in $C = \{(1,1),\,(1,3),\,(3,1),\,(3,3)\}$. Is there any case in which $(x, y), (y, z)$, but it doesn't follow that $(x, z)$? NO. So we conclude $C$ is transitive.
So $C$ is not reflexive, it is symmetric, and it is transitive, and
$D$ is not reflexive, not symmetric, not transitive.