# An example of preferences over a countable set that cannot be represented by a utility function

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?

• What countable means? It can be finite countable? Or only infinite countable? Commented Apr 13, 2019 at 11:23
• Both cases u can take Commented Apr 13, 2019 at 11:25
• I'll give you a tip first, and if you still can't solve the problem, comment later on. Commented Apr 13, 2019 at 11:26
• Ok.thnx fr the tips in advance. Commented Apr 13, 2019 at 11:27

If there is a function $$u: X \rightarrow \mathbb{R}$$ that represents the preference relation $$\succsim$$, then $$\succsim$$ is rational.
If $$\succsim$$ isn't rational, then all functions $$u$$ don't represent $$\succsim$$.