Math Problem Statement
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 >?
Solution
Let's break down the problem into two parts:
1. Case 1: is Finite or Countable
Given:
- is a complete and transitive binary relation over .
- is either finite or countable.
Claim: Any complete and transitive binary relation over can be represented by a utility function .
Proof:
-
Finite Case: If is finite, any complete and transitive binary relation can be represented by a utility function, as discussed previously. You can enumerate the elements of in order of the relation and assign increasing real numbers as utilities.
-
Countable Case: If is countable, similar reasoning applies. The completeness and transitivity of ensure that 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: and is Continuous, Complete, and Transitive
Given:
- (the set of all non-negative real numbers, excluding 0, often denoted or ).
- is a complete, transitive, and continuous binary relation over .
Claim: Is it possible to find a continuous utility function representing ?
Explanation:
-
Continuity: The continuity of means that if a sequence converges to , 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 ), there exists a continuous utility function that represents .
Thus, in this case, it is possible to find a continuous utility function that represents the relation .
Conclusion:
- For finite or countable , a utility function can always represent a complete and transitive binary relation.
- For , if the relation is also continuous, then there exists a continuous utility function representing .
Follow-up Questions:
- How does the result change if the space is uncountable but not continuous?
- What happens if is or higher-dimensional, with a continuous relation ?
- Can you construct a specific example of a continuous utility function for ?
- How does the assumption of continuity influence the possibility of constructing a utility function?
- 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.
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Math Problem Analysis
Mathematical Concepts
Binary Relations
Utility Functions
Complete and Transitive Relations
Continuous Functions
Debreu Representation Theorem
Formulas
-
Theorems
Debreu Representation Theorem
Suitable Grade Level
Advanced Undergraduate
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