<p>A relation that is not asymmetric need not be symmetric. But if <span class="math-container">$\succsim$</span> is not asymmetric, there must be elements <span class="math-container">$y$</span> and <span class="math-container">$x$</span> such that both <span class="math-container">$x\succsim y$</span> and <span class="math-container">$y\succsim x$</span> hold and these elements are all you need to arrive at a contradiction. </p>

<p>Also, <span class="math-container">$x\succsim y\implies y\succsim x$</span> can hold for an asymmetric relation, namely when <span class="math-container">$x\succsim y$</span> does not hold. </p>

<p>I'd like also to point out that "any" is ambiguous here and might be read as "all" instead
of the intended "some."</p>

CC BY-SA 4.0
2018-09-28T08:52:48.720Z
Ok I undestand that I need to improve the form, but is saying "for any" wrong? I intended it as ∀, why should I say "some"? I just want to be as precise as possible.
plastico
CC BY-SA 4.0
2018-09-28T09:43:31.713Z
I don't think it is wrong, but "some" is clearer in this context.
Michael Greinecker
CC BY-SA 4.0
2018-09-28T00:17:21.367Z
2018-09-28T00:17:21.367Z
Michael Greinecker
<h2>If <span class="math-container">$\succsim$</span> is transitive but irreflexive, then it is asymmetric.</h2>

<p>this is my proof:  </p>

<p>Suppose <span class="math-container">$\succsim$</span> is not asymmetric, which means that for any <span class="math-container">$x,y \in X$</span> <span class="math-container">$x\succsim y \rightarrow y \succsim x$</span>. By definition <span class="math-container">$\succsim$</span> is transitive, i.e. for any <span class="math-container">$x, y, z \in X$</span> we have <span class="math-container">$x \succsim y$</span> &amp; <span class="math-container">$y\succsim z$</span> <span class="math-container">$\rightarrow$</span> <span class="math-container">$x \succsim z$</span>. So since we are supposing <span class="math-container">$\succsim$</span> is symmetric:  <span class="math-container">$y\succsim x \rightarrow x \succsim$</span> y. Now since  <span class="math-container">$\succsim$</span>  is transitive  <span class="math-container">$y \succsim x$</span> &amp;  <span class="math-container">$x\succsim y \rightarrow y \succsim y$</span> but  <span class="math-container">$\succsim$</span>  is irreflexive so it's a contradition and  <span class="math-container">$\succsim$</span> is asymmetric.</p>

<p>What do you think of my proof, is it wrong or not clear enough?</p>

CC BY-SA 4.0
2018-09-27T21:20:18.363Z
Your proof looks good to me.
Herr K.
<preferences><self-study>
If $\succsim$ is transitive but irreflexive, then it is asymmetric, proof
plastico
# If $\succsim$ is transitive but irreflexive, then it is asymmetric, proof

(Asked by: username_0 on 2018-09-27)

## If $\succsim$ is transitive but irreflexive, then it is asymmetric.

this is my proof:

Suppose $\succsim$ is not asymmetric, which means that for any $x,y \in X$ $x\succsim y \rightarrow y \succsim x$. By definition $\succsim$ is transitive, i.e. for any $x, y, z \in X$ we have $x \succsim y$ & $y\succsim z$ $\rightarrow$ $x \succsim z$. So since we are supposing $\succsim$ is symmetric: $y\succsim x \rightarrow x \succsim$ y. Now since $\succsim$ is transitive $y \succsim x$ & $x\succsim y \rightarrow y \succsim y$ but $\succsim$ is irreflexive so it's a contradition and $\succsim$ is asymmetric.

What do you think of my proof, is it wrong or not clear enough?

**Tags:** preferences, self-study
> **Comments:**
> **username_1**: Your proof looks good to me.  

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## 1 answers

### Answer by username_2 on 2018-09-28. Score: 3 | Accepted answer

A relation that is not asymmetric need not be symmetric. But if $\succsim$ is not asymmetric, there must be elements $y$ and $x$ such that both $x\succsim y$ and $y\succsim x$ hold and these elements are all you need to arrive at a contradiction.

Also, $x\succsim y\implies y\succsim x$ can hold for an asymmetric relation, namely when $x\succsim y$ does not hold.

I'd like also to point out that "any" is ambiguous here and might be read as "all" instead
of the intended "some."

> **Comments:**
> **username_0**: Ok I undestand that I need to improve the form, but is saying "for any" wrong? I intended it as ∀, why should I say "some"? I just want to be as precise as possible.  
> **username_2**: I don't think it is wrong, but "some" is clearer in this context.  

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