Let's work through both parts of this problem from the image.

Part (a): Neither > nor ∼ is complete

In the context of preference or order relations:

1. Completeness means that for every pair of elements $x^1$ and $x^2$ in the set $X$, one of the following must hold:

   • $x^1 > x^2$ (i.e., $x^1$ is strictly preferred to $x^2$),
   • $x^2 > x^1$ (i.e., $x^2$ is strictly preferred to $x^1$), or
   • $x^1 \sim x^2$ (i.e., $x^1$ is indifferent to $x^2$).

   To argue that neither $>$ nor $\sim$ is complete, consider the following:

   • The strict preference relation $>$ is not complete because there could be pairs of elements where neither $x^1 > x^2$ nor $x^2 > x^1$ holds. For instance, if the two elements are equally preferred, neither is strictly better than the other.
   • The indifference relation $\sim$ is also not complete because, for some pairs $x^1$ and $x^2$, it may hold that neither $x^1 > x^2$ nor $x^1 \sim x^2$, but rather $x^2 > x^1$.

Thus, neither relation covers all possible pairs, implying that neither $>$ nor $\sim$ is complete.

Part (b): For any $x^1$ and $x^2$ in $X$, only one of the following holds: $x^1 > x^2$, or $x^2 > x^1$, or $x^1 \sim x^2$

This part asserts that the preference or order relation on the set $X$ is asymmetric and trichotomous, meaning for any pair $x^1$ and $x^2$:

• $x^1 > x^2$ (strict preference),
• $x^2 > x^1$ (strict reverse preference), or
• $x^1 \sim x^2$ (indifference).

To justify this:

• Asymmetry of the strict preference relation $>$ means that if $x^1 > x^2$, it cannot simultaneously be the case that $x^2 > x^1$.
• Trichotomy implies that one and only one of these relations must hold for any pair $x^1$ and $x^2$.

This means that there is no situation where none of these relations hold between two elements in $X$; one of the three must always be true.

Would you like further details on any part of the explanation? Here are some additional questions to explore these concepts:

1. What is the difference between a complete and incomplete relation?
2. How do we define trichotomy in the context of preference relations?
3. Can a relation be both asymmetric and transitive? Provide examples.
4. How can an incomplete relation affect decision-making processes?
5. Can you think of an example where two elements $x^1$ and $x^2$ are incomparable under a given preference relation?

Tip:

To better understand completeness, imagine ranking movies. If for any two movies, you can either prefer one over the other or be indifferent, your preference relation is complete. If sometimes you're unsure or the movies are incomparable, it's incomplete.