0

enter image description here

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

Ayan Bhowmick
  • 3
  • 2
  • 1
    Can you please some background and clarify what exactly you don't understand. – BrsG Aug 26 '21 at 21:24
  • 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 Aug 26 '21 at 22:05

1 Answers1

1

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.)

VARulle
  • 6,805
  • 10
  • 25
  • 1
    Can you not also argue that the relationship is not even defined if the difference is not a prime number? – BrsG Aug 27 '21 at 09:58
  • @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 Aug 28 '21 at 10:49
  • @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 Aug 28 '21 at 12:01
  • @BrsG, I think that's what I did here! – VARulle Aug 28 '21 at 19:20
  • 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 Aug 28 '21 at 19:51
  • @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 Aug 29 '21 at 21:40