<p>How do I show that considering the preference relation <span class="math-container">$\succsim$</span>, then  <span class="math-container">$\succ$</span> is not complete?</p>

<p>I tried the following (which I don't know if it's right) but I'd also like to know if it's possible to generalize and say that <span class="math-container">$\succ$</span> will never be complete.</p>

<p>Suppose <span class="math-container">$X = \{a,b\}$</span> with preference relation <span class="math-container">$\{ a \succsim a, a \succsim b, b \succsim b\}$</span>. Then, we <span class="math-container">$\succ$</span> can't be a complete relation on <span class="math-container">$X$</span> because neither <span class="math-container">$b \succsim a$</span> nor <span class="math-container">$b \not{\succsim} a$</span> are in <span class="math-container">$X$</span> and therefore it's impossible to conclude that either <span class="math-container">$a \succ b$</span> or <span class="math-container">$b \succ a$</span> is true.</p>

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2020-05-13T01:04:45.710Z
Thanks for the input! Is there a set $X$ such that $\succ$ is complete on it? For instance, would $X = \{a,b\}$ with preferences $\{ a \succsim a, a \succsim b, b \succsim b, b \not{\succsim} a\}$ be a valid set for which $\succ$ is complete?
Pedro Cunha
<preferences>
Completeness of the strict binary relation
Pedro Cunha
# Completeness of the strict binary relation

(Asked by: username_0 on 2020-05-13)

How do I show that considering the preference relation $\succsim$, then $\succ$ is not complete?

I tried the following (which I don't know if it's right) but I'd also like to know if it's possible to generalize and say that $\succ$ will never be complete.

Suppose $X = \{a,b\}$ with preference relation $\{ a \succsim a, a \succsim b, b \succsim b\}$. Then, we $\succ$ can't be a complete relation on $X$ because neither $b \succsim a$ nor $b \not{\succsim} a$ are in $X$ and therefore it's impossible to conclude that either $a \succ b$ or $b \succ a$ is true.

**Tags:** preferences
> **Comments:**
> **username_0**: Thanks for the input! Is there a set $X$ such that $\succ$ is complete on it? For instance, would $X = \{a,b\}$ with preferences $\{ a \succsim a, a \succsim b, b \succsim b, b \not{\succsim} a\}$ be a valid set for which $\succ$ is complete?  

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