To analyze the relation $R_5$ on the set $A = \{0, 1, 2, 3\}$, we will check the definitions of irreflexive, asymmetric, and intransitive.

Given Relation $R_5$

$R_5 = \{(0, 0), (0, 1), (0, 2), (1, 2)\}$

1. Irreflexive

A relation is irreflexive if no element relates to itself, meaning for every $x \in A$, $x R x$ must be false.

• For $0$: $(0, 0) \in R_5$ (not irreflexive)
• For $1$: $(1, 1) \notin R_5$
• For $2$: $(2, 2) \notin R_5$
• For $3$: $(3, 3) \notin R_5$

Since $(0, 0) \in R_5$, R5 is not irreflexive.

2. Asymmetric

A relation is asymmetric if whenever $x R y$, then $y R x$ must be false.

• Check pairs:
  • $(0, 1)$: $(1, 0) \notin R_5$ (holds)
  • $(0, 2)$: $(2, 0) \notin R_5$ (holds)
  • $(1, 2)$: $(2, 1) \notin R_5$ (holds)
  • $(0, 0)$: self-pair (asymmetry applies)

Since all pairs satisfy asymmetry, R5 is asymmetric.

3. Intransitive

A relation is intransitive if whenever $x R y$ and $y R z$, then $x R z$ must be false.

• Possible pairs:
  • From $(0, 1)$ and $(1, 2)$: $(0, 2) \in R_5$ (violates intransitivity)

Since $(0, 1)$ and $(1, 2)$ lead to $(0, 2)$ also being in $R_5$, R5 is not intransitive.

Conclusion

• R5 is not irreflexive.
• R5 is asymmetric.
• R5 is not intransitive.

Final Answer: R5 is asymmetric.

Do you want more details or have any questions? Here are some follow-up questions you might consider:

1. What are other examples of asymmetric relations?
2. How can we prove a relation is transitive?
3. Can a relation be both irreflexive and asymmetric?
4. What are the implications of a relation being intransitive?
5. How can we represent relations graphically?

Tip: Always check definitions carefully to clarify the properties of a relation.