To address the question of whether any complete and transitive binary relation $>$ over a set $X$ can be represented by a utility function under the conditions that $X$ is finite or countable, let's analyze this in two parts:

1. Finite Set $X$:

If $X$ is finite, then any complete and transitive binary relation $>$ over $X$ can indeed be represented by a utility function. This follows from the fact that for any finite set, you can assign a unique utility value to each element of $X$ in such a way that the utility function preserves the ordering defined by $>$.

Proof:

• Since $X$ is finite, enumerate its elements as $x_1, x_2, \dots, x_n$.
• Since $>$ is complete and transitive, the elements can be ordered such that $x_1 > x_2 > \dots > x_n$.
• Define a utility function $u: X \rightarrow \mathbb{R}$ by assigning increasing utility values to the elements according to this order: $u(x_1) > u(x_2) > \dots > u(x_n)$.
• This utility function $u$ represents the relation $>$ because it preserves the order: $x_i > x_j$ if and only if $u(x_i) > u(x_j)$.

Thus, the claim holds when $X$ is finite.

2. Countable Set $X$:

If $X$ is countable, the result can be more subtle. For a countable set $X$, any complete and transitive binary relation $>$ can also be represented by a utility function.

Explanation:

• For countable sets, the situation is similar to the finite case, but with the caveat that the utility function might need to take values in $\mathbb{R}$ (or another dense subset of the reals).
• The key result here is the Debreu Representation Theorem, which states that if $>$ is a complete, transitive, and continuous binary relation on a separable metric space (like $\mathbb{R}$ or $\mathbb{Q}$), then there exists a continuous utility function that represents $>$.

Since $X$ is countable, you can still enumerate the elements of $X$ as $x_1, x_2, \dots$. Under the completeness and transitivity conditions, it’s possible to construct a utility function $u: X \rightarrow \mathbb{R}$ that represents $>$.

Conclusion: The result is true for both finite and countable sets $X$. In both cases, a utility function can represent the complete and transitive binary relation $>$ over $X$.

Summary

• Finite $X$: The relation $>$ can be represented by a utility function.
• Countable $X$: The relation $>$ can still be represented by a utility function.

Follow-up Questions:

1. How would the situation change if $X$ were uncountable?
2. What if the binary relation $>$ was not transitive? Could a utility function still represent it?
3. How does the concept of a utility function change in higher-dimensional spaces?
4. Can you provide an example of constructing a utility function for a specific countable set and binary relation?
5. How does the continuity of the utility function influence the representation of the binary relation?

Tip:

When proving the existence of a utility function, remember that the transitivity and completeness of the binary relation are crucial. These properties ensure that the relation can be well-ordered, which is necessary for defining a corresponding utility function.