[SOLVED] Equivalence Relation

**dwsmith** · April 29th 2010, 01:13 PM

$a,b,c\in\mathbb{Z}$ $a\sim b$ iff. $2|(a+b)$

This is suppose to be an equivalence relation; however, when I did the transitivity, it didn't work out.

reflexive: $a\sim a$
$2|(a+a)\rightarrow 2p=2a$ Yes reflexive.

symmetric: $a\sim b$ then $b\sim a$
$2|(a+b) \rightarrow 2n=a+b\rightarrow 2n=b+a\rightarrow 2n|(b+a)$

Thus, $b\sim a$

transitive:
$a\sim b$ , $b\sim c$ , then $a\sim c$
$a\sim b\rightarrow 2m=a+b$
$b\sim c\rightarrow 2r=b+c$

Subtracted the equations:
$2(m-r)=a-c$

Again, I am supposed to conclude transitive what is going wrong here?

**Tikoloshe** · April 29th 2010, 01:41 PM

So how about $a+2b+c=2m+2n\implies a+c=2(m+n-b)$ ?

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