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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

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    $\begingroup$ Can you please some background and clarify what exactly you don't understand. $\endgroup$
    – BrsG
    Aug 26 at 21:24
  • $\begingroup$ 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 $\endgroup$
    – 1muflon1
    Aug 26 at 22:05
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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.)

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    $\begingroup$ Can you not also argue that the relationship is not even defined if the difference is not a prime number? $\endgroup$
    – BrsG
    Aug 27 at 9:58
  • $\begingroup$ @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. $\endgroup$
    – VARulle
    Aug 28 at 10:49
  • $\begingroup$ @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. $\endgroup$
    – BrsG
    Aug 28 at 12:01
  • $\begingroup$ @BrsG, I think that's what I did here! $\endgroup$
    – VARulle
    Aug 28 at 19:20
  • $\begingroup$ 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. $\endgroup$
    – BrsG
    Aug 28 at 19:51

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