$\displaystyle \forall a,b,c\in\mathbb{Z}$ $\displaystyle a\sim b$ iif. $\displaystyle \left\vert a-b \right\vert\leq 3$
Not sure how to show this. I know $\displaystyle a\sim b, b\sim c$, then $\displaystyle a\sim c$
Hello, dwsmith!
If I read the problem correctly, the relation is not transitive.
$\displaystyle \forall\: a,b,c\in\mathbb{Z}:\; a\sim b \:\text{ iff }|a-b| \:\leq\:3$
Not sure how to show this. I know $\displaystyle a\sim b, b\sim c$, then $\displaystyle a\sim c$
$\displaystyle a\sim b\,\text{ means: }\,a\text{ and }b\text{ are within 3 units of each other.}$
$\displaystyle b\sim c\,\text{ means: }\,b\text{ and }c\text{ are within 3 units of each other.}$
$\displaystyle \text{But this does }not\text{ imply }\,a\sim c,$
. . $\displaystyle \text{ that }a\text{ and }c\text{ are within 3 units of each other.}$