Consider S={1,2,3}S={1,2,3} and the following relation RR on SS:
R={⟨1,2⟩,⟨2,3⟩,⟨3,1⟩,⟨1,1⟩}R={⟨1,2⟩,⟨2,3⟩,⟨3,1⟩,⟨1,1⟩}. why is it reflexive

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.

Consider S = {1, 2, 3} and the following relation R on S: R = {⟨1, 2⟩, ⟨2, 3⟩, ⟨3, 1⟩, ⟨1, 1⟩}. Why is it reflexive?

Explore the concept of reflexivity in set theory with this example: S = {1, 2, 3} and relation R = {⟨1, 2⟩, ⟨2, 3⟩, ⟨3, 1⟩, ⟨1, 1⟩}. Understand why the relation is not reflexive due to missing pairs, and learn how to verify reflexivity in relations.

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