Finding relations on a set

Question

For each of the eight subsets of $\{ \mathrm{reflexive, symmetric, transitive} \}$, find a relation on $\{1,2,3\}$ that has the properties in that subset, but not the properties that are not in the subset.

Here is what I believe works for each. Let us denote the $8$ subsets by $1 = \emptyset, 2= \{r\}, 3 = \{s\}, 4 = \{t\}, 5 = \{r,s\}, 6 = \{r,t\}, 7 = \{s,t\}, 8 = \{r,s,t\}$.

Let $\rho_i$ be a relation with the properties of set $i$.

Then $\rho_1 = \emptyset$, $\rho_2 = \{(1,1)\}$, $\rho_3 = \{(1,2),(2,1)\}$, $\rho_4 = \{(1,2),(2,3),(1,3)\}$, $\rho_5 = \{(1,1), (1,2), (2,1)\} = \rho_1 \cup \rho_2$, $\rho_6 = \{(1,1), (1,2), (2,3), (1,3)\} = \rho_1 \cup \rho_4$, $\rho_7 = \{(1,2),(2,1), (2,3),(1,3)\} = \rho_3 \cup \rho_4$, and $\rho_8 = \{(1,1), (1,2), (2,1), (2,3), (1,3), (3,2), (3,1)\}$, the last one, somewhat surprisingly to me, is not $\rho_2 \cup \rho_3 \cup \rho_4$. My question is, do these $\rho_i$ work? For example in the only reflexive set I do not have $(2,2)$, but I believe this is fine, is it?

Answer 1

1. $\rho_1$ is actually symmetric and transitive. Both properties hold vacuously as no elements are related to each other.
2. $\rho_2$ is not reflexive. You require $(2,2)$ and $(3,3)$ to be inside the relation as well. It is also symmetric and transitive.
3. $\rho_3$ is good.
4. $\rho_4$ is good.
5. $\rho_5$, similar to $\rho_2$, requires $(2,2)$ and $(3,3)$ to be in the relation for it to be reflexive. It is also transitive.
6. $\rho_6$ has similar issues on reflexivity.
7. $\rho_7$ is not symmetric.
8. $\rho_8$ has similar issues in reflexivity. For this, you can simply take the power set.

Can you work out the issues I've listed out?