Prove or provide counter-example: any complete and transitive binary relation > over X can be represented by a utility function, if: e e In X X (1), jis is if. finite; countable. [2+5] X = R* and & is continuous over X in addition to complete- ness and transitivity, is it possible to find a continuous utility function representing >?

Let's break down the problem into two parts:

### 1. **Case 1: \( X \) is Finite or Countable**

Given:
- \( > \) is a complete and transitive binary relation over \( X \).
- \( X \) is either finite or countable.

**Claim**: Any complete and transitive binary relation \( > \) over \( X \) can be represented by a utility function \( u: X \to \mathbb{R} \).

**Proof**:

- **Finite Case**: If \( X \) is finite, any complete and transitive binary relation can be represented by a utility function, as discussed previously. You can enumerate the elements of \( X \) in order of the relation \( > \) and assign increasing real numbers as utilities.

- **Countable Case**: If \( X \) is countable, similar reasoning applies. The completeness and transitivity of \( > \) ensure that \( X \) can be well-ordered according to \( > \). This allows us to assign utility values that preserve the order.

In both cases, a utility function can represent \( > \).

### 2. **Case 2: \( X = \mathbb{R}^* \) and \( > \) is Continuous, Complete, and Transitive**

Given:
- \( X = \mathbb{R}^* \) (the set of all non-negative real numbers, excluding 0, often denoted \( \mathbb{R}^+ \) or \( \mathbb{R}^* \)).
- \( > \) is a complete, transitive, and continuous binary relation over \( X \).

**Claim**: Is it possible to find a continuous utility function \( u: X \to \mathbb{R} \) representing \( > \)?

**Explanation**:

- **Continuity**: The continuity of \( > \) means that if a sequence \( x_n \) converges to \( x \), then the ordering of the elements in the sequence should reflect in the utility values smoothly.

- **Debreu's Theorem**: According to the **Debreu Representation Theorem**, if \( > \) is a continuous, complete, and transitive binary relation on a topological space (like \( \mathbb{R}^* \)), there exists a continuous utility function \( u: X \to \mathbb{R} \) that represents \( > \).

Thus, in this case, it **is possible** to find a continuous utility function that represents the relation \( > \).

### Conclusion:
- For finite or countable \( X \), a utility function can always represent a complete and transitive binary relation.
- For \( X = \mathbb{R}^* \), if the relation \( > \) is also continuous, then there exists a continuous utility function representing \( > \).

### Follow-up Questions:
1. How does the result change if the space \( X \) is uncountable but not continuous?
2. What happens if \( X \) is \( \mathbb{R}^2 \) or higher-dimensional, with a continuous relation \( > \)?
3. Can you construct a specific example of a continuous utility function for \( X = \mathbb{R}^* \)?
4. How does the assumption of continuity influence the possibility of constructing a utility function?
5. What if \( > \) is not continuous—can a utility function still be constructed?

### Tip:
When dealing with relations on real spaces, continuity is a powerful assumption that allows the application of theorems like Debreu's to ensure the existence of utility functions.

Explore how complete and transitive binary relations over finite, countable sets, and \( \mathbb{R}^* \) can be represented by utility functions. Learn about the conditions under which a continuous utility function exists for continuous, complete, and transitive relations on \( \mathbb{R}^* \).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation