Ok. I have a very hard question. Hopefully someone here can solve it. Now for binary operations, well I have one here, and I need to know if there exists in R (the set of real numbers), an identity element for the operation *. Here is the operation, x*y=abs(x-y). Now when I do it, I break up the absolute value into x-e for x-e greater than 0 and -(x-e) for x-e less than 0, where e is the identity element. Please explain your steps to this problem.

Question

anonymous · Answer

x = 1; y= 1/2

mathmate · Answer

You are looking for e such that x*e=abs(x-e) $ \forall x$.
Since x*y maps to only the positive half of R, there is no such identity element in $ R$.
However, e exists in $ R^+ $.