Consider S={1,2,3} and the following relation R on S:
R={⟨1,2⟩,⟨2,3⟩,⟨3,1⟩,⟨1,1⟩,⟨2,2⟩,⟨3,3⟩}.
Pick the right properties of R


:


Reflexive


Symmetric


Transitive


Total


Irreflexive:   ∀x : ⟨x,x⟩∉R


Asymmetric:   ∀x,y : ⟨x,y⟩∈R → ⟨y,x⟩∉R


Anti-symmetric


Intransitive:   ∀x,y,z : (⟨x,y⟩∈R ∧ ⟨y,z⟩∈R) → ⟨x,z⟩∉R

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 \(

Consider S={1,2,3} and the following relation R on S: R={⟨1,2⟩,⟨2,3⟩,⟨3,1⟩,⟨1,1⟩,⟨2,2⟩,⟨3,3⟩}. Pick the right properties of R: Reflexive, Symmetric, Transitive, Total, Irreflexive, Asymmetric, Anti-symmetric, Intransitive.

Analyze the properties of a relation R on the set S={1,2,3} and determine if it is reflexive, symmetric, transitive, and more. This problem covers key concepts from set theory and relations. Suitable for students in Grades 9-12.

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