Let's break down the given properties one by one to determine whether the relation $R$ defined as $R = \{ \langle x, y \rangle : x, y \in \mathbb{N}^+, x = y \}$ satisfies them.

1. Irreflexive:

• A relation $R$ is irreflexive if $\forall x : \langle x, x \rangle \notin R$.
• However, by the definition of $R$, $\langle x, x \rangle \in R$ for all $x \in \mathbb{N}^+$ since the relation only contains pairs where $x = y$.
• Conclusion: The relation is not irreflexive because $\langle x, x \rangle \in R$ for all $x$.

2. Asymmetric:

• A relation $R$ is asymmetric if $\forall x, y : \langle x, y \rangle \in R \rightarrow \langle y, x \rangle \notin R$.
• In the case of $R$, we only have pairs where $x = y$, meaning $\langle x, y \rangle = \langle y, x \rangle$, so symmetry is always present when $x = y$.
• Conclusion: The relation is not asymmetric because $\langle x, y \rangle \in R$ implies $\langle y, x \rangle \in R$ when $x = y$.

3. Anti-symmetric:

• A relation $R$ is anti-symmetric if $\forall x, y : (\langle x, y \rangle \in R \wedge \langle y, x \rangle \in R) \rightarrow x = y$.
• In $R$, the only pairs that exist are $\langle x, x \rangle$, which trivially satisfies the anti-symmetry condition because if $\langle x, y \rangle \in R$ and $\langle y, x \rangle \in R$, then $x = y$.
• Conclusion: The relation is anti-symmetric.

4. Intransitive:

• A relation $R$ is intransitive if $\forall x, y, z : (\langle x, y \rangle \in R \wedge \langle y, z \rangle \in R) \rightarrow \langle x, z \rangle \notin R$.
• Since $R$ only includes pairs where $x = y$, this condition doesn't hold because if $\langle x, y \rangle \in R$ and $\langle y, z \rangle \in R$, then $x = y = z$, and thus $\langle x, z \rangle \in R$.
• Conclusion: The relation is not intransitive because the transitivity condition holds.

Final Summary:

• The relation $R$ is:
  • Not irreflexive
  • Not asymmetric
  • Anti-symmetric
  • Not intransitive

Would you like more details or clarifications?

Here are 5 related questions to expand on the topic:

1. What is the difference between asymmetric and anti-symmetric relations?
2. Can a relation be both reflexive and anti-symmetric?
3. How do transitive and intransitive relations differ in mathematical terms?
4. What are examples of irreflexive relations in real-world applications?
5. Can a relation be symmetric and anti-symmetric at the same time?

Tip: Anti-symmetric relations allow pairs $\langle x, y \rangle$ and $\langle y, x \rangle$ only when $x = y$, so reflexive relations can still be anti-symmetric.