1. ## [SOLVED] Transitive

$\displaystyle \forall a,b,c\in\mathbb{Z}$ $\displaystyle a\sim b$ 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 $\displaystyle a\sim c$

2. Originally Posted by dwsmith
$\displaystyle \forall a,b,c\in\mathbb{Z}$ $\displaystyle a\sim b$ 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 $\displaystyle a\sim c$
it is not transitive

$\displaystyle \mid 5 - 3 \mid \leq 3$

$\displaystyle \mid 7-5 \mid \leq 3$

but

$\displaystyle \mid 7 - 3 \mid = 4 > 3$

counter example

3. 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 $\displaystyle a\sim c$

$\displaystyle a\sim b\,\text{ means: }\,a\text{ and }b\text{ are within 3 units of each other.}$

$\displaystyle b\sim c\,\text{ means: }\,b\text{ and }c\text{ are within 3 units of each other.}$

$\displaystyle \text{But this does }not\text{ imply }\,a\sim c,$
. . $\displaystyle \text{ that }a\text{ and }c\text{ are within 3 units of each other.}$