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Give an example of preferences over a countable set in which the preferences cannot be represented by a utility function that returns only integers as values.

I know a utility function exist that represents preference over a countable set. I know a few types of preferences such as lexicographic, monotonic, quasi linear. I have always dealt with utility function assuming preferences are represented by that utility function. So how to solve this question?

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  • $\begingroup$ What countable means? It can be finite countable? Or only infinite countable? $\endgroup$ – Marcelo Gelati Apr 13 at 11:23
  • $\begingroup$ Both cases u can take $\endgroup$ – panda Apr 13 at 11:25
  • $\begingroup$ I'll give you a tip first, and if you still can't solve the problem, comment later on. $\endgroup$ – Marcelo Gelati Apr 13 at 11:26
  • $\begingroup$ Ok.thnx fr the tips in advance. $\endgroup$ – panda Apr 13 at 11:27
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There is a proposition that says:

If there is a function $u: X \rightarrow \mathbb{R}$ that represents the preference relation $\succsim$, then $\succsim$ is rational.

So, the contrapositive form of this statement is:

If $\succsim$ isn't rational, then all functions $u$ don't represent $\succsim$.

So, it's a good approach to create a preference relation that isn't rational, i.e., the preference must violate transitivity or completeness.

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