A relation $R$ on a set $S$ is said to be reflexive if, for every element $x \in S$, the pair $\langle x, x \rangle$ is included in the relation $R$. In other words, each element of the set must be related to itself.

Let’s check if the relation $R$ is reflexive on the set $S = \{1, 2, 3\}$.

The relation $R$ is given as: $R = \{\langle 1, 2 \rangle, \langle 2, 3 \rangle, \langle 3, 1 \rangle, \langle 1, 1 \rangle\}$

To be reflexive, we need to verify that for each element of $S$, the pair $\langle x, x \rangle$ is in $R$:

• For $x = 1$, we have $\langle 1, 1 \rangle \in R$, which is true.
• For $x = 2$, we do not have $\langle 2, 2 \rangle \in R$.
• For $x = 3$, we do not have $\langle 3, 3 \rangle \in R$.

Since the pairs $\langle 2, 2 \rangle$ and $\langle 3, 3 \rangle$ are missing from $R$, the relation is not reflexive.

If the question intended to ask why the relation isn't reflexive, the answer would be that the pairs $\langle 2, 2 \rangle$ and $\langle 3, 3 \rangle$ are missing.

Would you like further clarification or details on reflexive relations?

Here are 5 related questions for further exploration:

1. What is a symmetric relation?
2. What is a transitive relation?
3. How do you check if a relation is an equivalence relation?
4. Can a relation be both reflexive and symmetric but not transitive?
5. How can you make the given relation $R$ reflexive?

Tip: Reflexivity requires every element to relate to itself, so always check for the presence of $\langle x, x \rangle$ for all elements in the set.