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Thread: [SOLVED] Transitive

  1. #1
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    [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 $\displaystyle a\sim c$
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by dwsmith View Post
    $\displaystyle \forall a,b,c\in\mathbb{Z}$ $\displaystyle a\sim b$ 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
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  3. #3
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    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 $\displaystyle a\sim c$

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

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


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

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