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Consider the operation ⊥ defined by placing, for every $x,y\in Z$
$x⊥y=x+|y|$, Check Associativity and Commutativity. Is there a Identity element in $Z$?
My proof:
Associativity
$x⊥(y⊥z)=(x⊥y)⊥z$
$x⊥(y⊥z)=x+|y+|z||$
$(x⊥y)⊥z=(x+|y|)+|z|=x+|y|+|z|\not=x+|y+|z||$
Commutativity
$x⊥y=y⊥x$
$x⊥y=x+|y|$
$y⊥x=y+|x|\not=x+|y|$
Identity element
$x⊥e=e⊥x=x$
$x⊥e=x+|e|=x\Rightarrow e=0?$
$e⊥x=0+|x|\not=x$
so no condition has been met?

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    $\begingroup$ You might want to use specific examples for $x, y,$ and $z$ to "witness" that equality does not in fact hold, but other than minor detail your work looks sound. $\endgroup$ – Robert Shore Oct 7 '19 at 21:34
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    $\begingroup$ I'd put that even stronger: you really should provide concrete counterexamples. $\endgroup$ – Bram28 Oct 7 '19 at 22:37
  • $\begingroup$ Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. $\endgroup$ – Shaun Oct 7 '19 at 23:52

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