Having trouble determining relation equivalences when there's a divisor involved. Here's an example question I'm trying to work out (where ~ is an equivalence relation).
When $X = \Bbb Z$, and a ~ b, there is an integer $p ≥ 4$ such that $p^2 | (a - b)$
The steps I'd have to take to establish whether or not it has a relation equivalence would be to determine if it's reflective, symmetric, and transitive. Where would I start with this one?
$\ a\ne b,\,a\sim b\,\Rightarrow |a-b| \ge 4^2.\,$ Hence we can disprove transitivity by finding some $\,a\sim c,\ c\sim b\,$ such that we have $|a-b| < 4^2,\ $ e.g. choose $\,a\,$ with $\ a\sim 0,\, 0\sim 16\,$ for $\,16 < a < 32.$
Is it reflexive?
Given $a\in\mathbb Z$, is it true that there is some integer $p\geq 4$ such that $p^2 | (a-a)$? Of course this is true because every positive integer divides $0$.
Is it symmetric?
Given $a,b\in\mathbb Z$ and $p\geq 4$ where $p^2|(a-b)$, is it true that there is some integer $q\geq 4$ such that $q^2 | (b-a)$? Certainly, just take $q=p$.
Is it transitive?
Given $a,b,c \in\mathbb Z$, and $p,q\geq 4$ where $p^2|(a-b)$ and $q^2|(b-c)$, is it true that there is some integer $r\geq 4$ such that $r^2|(a-c)$? Unfortunately, no. As a counterexample, take $a=41, b=25, c=0, p=4, q=5$. Then $p^2|(a-b)$ and $q^2|(b-c)$, but unfortunately, there is no integer $r$ such that $r^2 |(a-c)$.
