Prove or provide counter-example: any complete and transitive binary 
relation > over X can be represented by a utility function, if: 
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finite; 
countable. 

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.

Explore how complete and transitive binary relations over finite or countable sets can be represented by utility functions. Understand the implications of the Debreu Representation Theorem in mathematical economics and decision theory.

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