<p>There is a proposition that says:</p>

<blockquote>
  <p>If there is a function <span class="math-container">$u: X \rightarrow \mathbb{R}$</span> that represents the preference relation <span class="math-container">$\succsim$</span>, then <span class="math-container">$\succsim$</span> is rational.</p>
</blockquote>

<p>So, the contrapositive form of this statement is:</p>

<blockquote>
  <p>If <span class="math-container">$\succsim$</span> isn't rational, then all functions <span class="math-container">$u$</span> don't represent <span class="math-container">$\succsim$</span>.</p>
</blockquote>

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

CC BY-SA 4.0
2020-02-24T00:46:58.667Z
Whatever your example is, it's probably wrong. The question assumes the relation is transitive and complete.
Guilherme Jardim Duarte
CC BY-SA 4.0
2019-04-13T11:38:55.797Z
2019-04-13T11:38:55.797Z
Marcelo Gelati
<p>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.</p>

<p>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? </p>

CC BY-SA 4.0
2019-04-13T11:23:12.860Z
What countable means? It can be finite countable? Or only infinite countable?
Marcelo Gelati
CC BY-SA 4.0
2019-04-13T11:25:12.740Z
Both cases u can take
panda
CC BY-SA 4.0
2019-04-13T11:26:47.807Z
I'll give you a tip first, and if you still can't solve the problem, comment later on.
Marcelo Gelati
CC BY-SA 4.0
2019-04-13T11:27:42.460Z
Ok.thnx fr the tips in advance.
panda
<preferences>
An example of preferences over a countable set that cannot be represented by a utility function
panda
# An example of preferences over a countable set that cannot be represented by a utility function

(Asked by: username_0 on 2019-04-13)

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?

**Tags:** preferences

> **Comments:**
> **username_1**: What countable means? It can be finite countable? Or only infinite countable?  
> **username_0**: Both cases u can take  
> **username_1**: I'll give you a tip first, and if you still can't solve the problem, comment later on.  
> **username_0**: Ok.thnx fr the tips in advance.  

---

## 1 answers

### Answer by username_1 on 2019-04-13. Score: 2

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.

> **Comments:**
> **username_2**: Whatever your example is, it's probably wrong. The question assumes the relation is transitive and complete.  

---

