Given the set $S = \{1, 2, 3\}$ and the relation $R = \{ \langle 1,2 \rangle, \langle 2,3 \rangle, \langle 3,1 \rangle, \langle 1,1 \rangle, \langle 2,2 \rangle, \langle 3,3 \rangle \}$, let's analyze the properties of the relation $R$ in detail.

Reflexive:

• A relation is reflexive if for every element $x \in S$, $\langle x, x \rangle \in R$.
• Here, $\langle 1,1 \rangle$, $\langle 2,2 \rangle$, and $\langle 3,3 \rangle$ are all in $R$, so $R$ is reflexive.

Symmetric:

• A relation is symmetric if for every pair $\langle x, y \rangle \in R$, we also have $\langle y, x \rangle \in R$.
• In $R$, we have $\langle 1, 2 \rangle$ but not $\langle 2, 1 \rangle$, and $\langle 2, 3 \rangle$ but not $\langle 3, 2 \rangle$, and similarly for $\langle 3, 1 \rangle$.
• Therefore, $R$ is not symmetric.

Transitive:

• A relation is transitive if whenever $\langle x, y \rangle \in R$ and $\langle y, z \rangle \in R$, then $\langle x, z \rangle \in R$.
• We can check:
  • $\langle 1, 2 \rangle \in R$ and (