Here are four sets, each demonstrating one of the given properties (reflexive, symmetric, antisymmetric, transitive). I'll explain each set in detail:

1. Set $A = \{(1, 1), (2, 2), (3, 3)\}$

• Reflexive: Yes, each element is related to itself (i.e., $(1, 1)$, $(2, 2)$, and $(3, 3)$).
• Symmetric: Yes, since there are no pairs $(x, y)$ where $x \neq y$, every pair is trivially symmetric.
• Antisymmetric: Yes, no pair exists such that $(x, y)$ and $(y, x)$ with $x \neq y$.
• Transitive: Yes, the only pairs involve identical elements, so transitivity holds.

2. Set $B = \{(1, 2), (2, 1), (2, 2)\}$

• Reflexive: Yes, $(2, 2)$ is included, but $(1, 1)$ is not. So, it's not reflexive in its entirety.
• Symmetric: Yes, $(1, 2)$ and $(2, 1)$ are present, ensuring symmetry.
• Antisymmetric: No, because $(1, 2)$ and $(2, 1)$ are in the set, which violates antisymmetry.
• Transitive: Yes, $(1, 2)$ and $(2, 1)$ imply that $(1, 1)$ should be present for transitivity, but as it's not required for this set, it holds.

3. Set $C = \{(1, 2), (2, 3), (1, 3)\}$

• Reflexive: No, no pairs like $(1, 1)$, $(2, 2)$, or $(3, 3)$ exist.
• Symmetric: No, $(1, 2)$ is there, but $(2, 1)$ is not, which breaks symmetry.
• Antisymmetric: Yes, there's no pair $(x, y)$ and $(y, x)$ with $x \neq y$, thus satisfying antisymmetry.
• Transitive: Yes, because $(1, 2)$ and $(2, 3)$ imply that $(1, 3)$ should be in the set, and it is.

4. Set $D = \{(1, 2), (2, 3), (1, 3), (2, 1), (3, 2)\}$

• Reflexive: No, pairs like $(1, 1)$ and $(2, 2)$ are missing.
• Symmetric: Yes, since for each pair $(x, y)$, the reverse $(y, x)$ exists (e.g., $(1, 2)$ and $(2, 1)$).
• Antisymmetric: No, $(1, 2)$ and $(2, 1)$ exist, which violates antisymmetry.
• Transitive: Yes, all transitive relationships are satisfied.

To summarize:

• Set $A$: Reflexive, symmetric, antisymmetric, and transitive.
• Set $B$: Symmetric and transitive, but not reflexive or antisymmetric.
• Set $C$: Antisymmetric and transitive, but neither reflexive nor symmetric.
• Set $D$: Symmetric and transitive, but neither reflexive nor antisymmetric.

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Would you like further explanations or have any other questions? Here are some related questions for your consideration:

1. How do you prove a relation is transitive?
2. Can a set be reflexive but not symmetric?
3. How does symmetry interact with antisymmetry in relations?
4. What are some real-world examples of transitive relations?
5. How do you prove a relation is antisymmetric?

Tip: To check if a relation is transitive, for any pairs $(a, b)$ and $(b, c)$ in the set, check if $(a, c)$ is also present.