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yes, but we found that e = 2x
and I showed that x = 0
so naturally e = 2x ---> e = 2*0 ----> e = 0

x * 0 = x for all x (when x > 0 or x < 0)

yes, sry i was trying a different way to do this, but not really finding anything

you want to find an identity element (e) for x * y = |x - y|
not for x * y = x - y

then we check to see if e = 0 is actually the true identity element for x * y = |x - y|

yes x > 0 means |x - 0| = x - 0, but we want something that will apply to both x > 0 and x < 0

So then there is no identity element for the set of real numbers for this operation.?

yes it looks like it since it doesn't apply to all real numbers

But what about x-e==x. That can have negative x values. And x>0. Is that what you are saying?

yes x - e = x can have negative values, but we're referring to |x-y| not x-y

this is why it's very important to check the possible solutions

Ok thank you so much.