Consider *S*={1,2,3} and the following relation *R* on *S*:
*R*={⟨1,1⟩,⟨1,2⟩,⟨1,3⟩,⟨2,1⟩,⟨3,1⟩}.
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*

Give the answer without explanation

The correct properties of the relation *R* are:

- Irreflexive
- Asymmetric
- Anti-symmetric
- Intransitive

Let me know if you'd like further clarification or have any questions!

Here are some related questions:

1. What does it mean for a relation to be reflexive or irreflexive?
2. How can you determine if a relation is symmetric or asymmetric?
3. What conditions must be met for a relation to be transitive?
4. Can a relation be both symmetric and anti-symmetric?
5. What does total relation mean in the context of a set?

**Tip:** A relation is asymmetric if whenever *x* is related to *y*, *y* is never related back to *x*.

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

Explore the properties of the relation *R* defined on the set *S*={1,2,3}. This problem discusses the concepts of reflexivity, symmetry, transitivity, and more. Suitable for students in Grades 10-12, it helps reinforce key topics in set theory and relations.

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