<p>First of all, despite the context and the symbol they use, forget about &quot;preferences&quot; and think of this as a purely abstract binary relation between numbers.</p>
<p>By definition, <em>completeness</em> would require that for all <span class="math-container">$x$</span> and <span class="math-container">$y$</span> in <span class="math-container">$X$</span> you either have <span class="math-container">$x\succsim y$</span> or <span class="math-container">$y\succsim x$</span> (or both). Remember that therefore finding a single counterexample already disproves completeness.</p>
<p>According to the definition of the relation <span class="math-container">$\succsim$</span> in the exercise, this means that for all <span class="math-container">$x$</span> and <span class="math-container">$y$</span> in <span class="math-container">$X$</span> either <span class="math-container">$x-y$</span> or <span class="math-container">$y-x$</span> has to be a positive prime number. Now try <span class="math-container">$x=10$</span> and <span class="math-container">$y=4$</span>, as in the answer. Is <span class="math-container">$10-4=6$</span> a positive prime number? Is <span class="math-container">$4-10=-6$</span> one?</p>
<p>(The choice of <span class="math-container">$10$</span> and <span class="math-container">$4$</span> in the answer is arbitrary. Choosing <span class="math-container">$x=y=1$</span> also works, as do thousands of other pairs of numbers.)</p>

CC BY-SA 4.0
2021-08-27T09:58:12.573Z
Can you not also argue that the relationship is not even *defined* if the difference is not a prime number?
BrsG
CC BY-SA 4.0
2021-08-28T10:49:16.197Z
@BrsG, well, the relation itself is defined as the set of all pairs of numbers for which the distance is a prime. So the relation is well defined. It's just that some pairs are not an element of this set, i.e. some pairs of numbers are not related.
VARulle
CC BY-SA 4.0
2021-08-28T12:01:35.927Z
@VarRulle: Indeed! But for completeness, doesn't it have to defined for *all* pairs in $\{1,\cdots, 100\}$? So couldn't you just used that as an argument that the relation is not complete.
BrsG
CC BY-SA 4.0
2021-08-28T19:20:17.567Z
@BrsG, I think that's what I did here!
VARulle
CC BY-SA 4.0
2021-08-28T19:51:41.793Z
Yes, implicitly. It's just about a slight nuance regarding the conclusion: "The relationship doesn't hold" vs "The relationship is not defined" (for 10 and 4). The counter example in the slides suggest it doesn't hold, which is misleading, because it is not even defined (for 10 and 4).  Your answer doesn't make this explicit, but still +1.
BrsG
CC BY-SA 4.0
2021-08-29T21:40:24.050Z
@BrsG, I shouldn't have written "a relation between numbers", as this might be the source of confusion. There is no "relationship" here. There is a relation, which is a set of pairs of numbers. If a pair $(x,y)$ is in the set, then one says that "$x$ is related to $y$". Now if $(x,y)$ is not in the set, one can say that "$x$ is not related to $y$", but one shouldn't say that "the relation between $x$ and $y$ is undefined", since this confuses the relation (the set) with the property of a pair of numbers being an element of the set. And what "doesn't hold" is completeness, not the relation.
VARulle
CC BY-SA 4.0
2021-08-27T09:43:20.823Z
2021-08-27T09:43:20.823Z
VARulle
<p><a href="https://i.sstatic.net/GNyon.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GNyon.png" alt="enter image description here" /></a></p>
<p>Given X = {1,2,..., 100}. For x, y in X, define x # y if and only if x - y is a positive prime number. Is the # relation incomplete? I don't particularly understand the reasoning as of yet, and though I have a general idea after looking up the difference between indifferent and incomplete, I can't quite put into words why this would work for a numbered set.</p>
<p>Background: we are going over preferences and what makes them complete or incomplete. Here, we define the &amp; symbol (the &gt;= sign in the image) as the preference equation. x is preferred to y if and only if x - y is a positive prime number; I don't understand why this preference is incomplete</p>

CC BY-SA 4.0
2021-08-26T21:24:49.003Z
Can you please some background and clarify what exactly you don't understand.
BrsG
CC BY-SA 4.0
2021-08-26T22:05:27.573Z
Please type out the text in pictures and use MathJax for equations, e.g. to write $x-\beta y =0$ you can write `$x-\beta y=0$` pictures should be reserved for plots and graphs as otherwise equations or text is not searchable
1muflon1
<preferences>
Is this Incomplete or Indifferent?
Ayan Bhowmick
# Is this Incomplete or Indifferent?

(Asked by: username_0 on 2021-08-26)

[![enter image description here](https://i.sstatic.net/GNyon.png)](https://i.sstatic.net/GNyon.png)

Given X = {1,2,..., 100}. For x, y in X, define x # y if and only if x - y is a positive prime number. Is the # relation incomplete? I don't particularly understand the reasoning as of yet, and though I have a general idea after looking up the difference between indifferent and incomplete, I can't quite put into words why this would work for a numbered set.

Background: we are going over preferences and what makes them complete or incomplete. Here, we define the & symbol (the >= sign in the image) as the preference equation. x is preferred to y if and only if x - y is a positive prime number; I don't understand why this preference is incomplete

**Tags:** preferences

> **Comments:**
> **username_1**: Can you please some background and clarify what exactly you don't understand.  
> **username_2**: Please type out the text in pictures and use MathJax for equations, e.g. to write $x-\beta y =0$ you can write `$x-\beta y=0$` pictures should be reserved for plots and graphs as otherwise equations or text is not searchable  

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## 1 answers

### Answer by username_3 on 2021-08-27. Score: 1 | Accepted answer

First of all, despite the context and the symbol they use, forget about "preferences" and think of this as a purely abstract binary relation between numbers.

By definition, *completeness* would require that for all $x$ and $y$ in $X$ you either have $x\succsim y$ or $y\succsim x$ (or both). Remember that therefore finding a single counterexample already disproves completeness.

According to the definition of the relation $\succsim$ in the exercise, this means that for all $x$ and $y$ in $X$ either $x-y$ or $y-x$ has to be a positive prime number. Now try $x=10$ and $y=4$, as in the answer. Is $10-4=6$ a positive prime number? Is $4-10=-6$ one?

(The choice of $10$ and $4$ in the answer is arbitrary. Choosing $x=y=1$ also works, as do thousands of other pairs of numbers.)

> **Comments:**
> **username_1**: Can you not also argue that the relationship is not even *defined* if the difference is not a prime number?  
> **username_3**: @username_1, well, the relation itself is defined as the set of all pairs of numbers for which the distance is a prime. So the relation is well defined. It's just that some pairs are not an element of this set, i.e. some pairs of numbers are not related.  
> **username_1**: @VarRulle: Indeed! But for completeness, doesn't it have to defined for *all* pairs in $\{1,\cdots, 100\}$? So couldn't you just used that as an argument that the relation is not complete.  
> **username_3**: @username_1, I think that's what I did here!  
> **username_1**: Yes, implicitly. It's just about a slight nuance regarding the conclusion: "The relationship doesn't hold" vs "The relationship is not defined" (for 10 and 4). The counter example in the slides suggest it doesn't hold, which is misleading, because it is not even defined (for 10 and 4).  Your answer doesn't make this explicit, but still +1.  
> **username_3**: @username_1, I shouldn't have written "a relation between numbers", as this might be the source of confusion. There is no "relationship" here. There is a relation, which is a set of pairs of numbers. If a pair $(x,y)$ is in the set, then one says that "$x$ is related to $y$". Now if $(x,y)$ is not in the set, one can say that "$x$ is not related to $y$", but one shouldn't say that "the relation between $x$ and $y$ is undefined", since this confuses the relation (the set) with the property of a pair of numbers being an element of the set. And what "doesn't hold" is completeness, not the relation.  

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