To determine whether each relation is a partial ordering, we need to check if the relation is:

1. Reflexive: Every element is related to itself, i.e., $(x, x)$ is in the relation for every $x$ in the set.
2. Antisymmetric: If $(x, y)$ and $(y, x)$ are in the relation, then $x = y$.
3. Transitive: If $(x, y)$ and $(y, z)$ are in the relation, then $(x, z)$ must also be in the relation.

The set in question is $\{0, 1, 2, 3\}$. Let's evaluate each relation.

a) $\{(0, 0), (2, 2), (3, 3)\}$

• Reflexive: Not reflexive because $(1, 1)$ is missing.
• Antisymmetric: Yes, all pairs are of the form $(x, x)$, so antisymmetry holds trivially.
• Transitive: Yes, since there are no pairs like $(x, y)$ and $(y, z)$ for $x \neq y$, transitivity holds trivially.

Conclusion: This is not a partial ordering because it is not reflexive.

b) $\{(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 3)\}$

• Reflexive: Not reflexive because $(3, 3)$ is present but $(1, 1)$ is missing.
• Antisymmetric: Yes, antisymmetry holds trivially_ans_parser