# operation on set proof

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?

• 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. – Robert Shore Oct 7 '19 at 21:34
• I'd put that even stronger: you really should provide concrete counterexamples. – Bram28 Oct 7 '19 at 22:37
• 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. – Shaun Oct 7 '19 at 23:52