1. ## [SOLVED] Transitive

$\forall a,b,c\in\mathbb{Z}$ $a\sim b$ iif. $\left\vert a-b \right\vert\leq 3$

Not sure how to show this. I know $a\sim b, b\sim c$, then $a\sim c$

2. Originally Posted by dwsmith
$\forall a,b,c\in\mathbb{Z}$ $a\sim b$ iif. $\left\vert a-b \right\vert\leq 3$

Not sure how to show this. I know $a\sim b, b\sim c$, then $a\sim c$
it is not transitive

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

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

but

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

counter example

3. Hello, dwsmith!

If I read the problem correctly, the relation is not transitive.

$\forall\: a,b,c\in\mathbb{Z}:\; a\sim b \:\text{ iff }|a-b| \:\leq\:3$

Not sure how to show this. I know $a\sim b, b\sim c$, then $a\sim c$

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

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

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