An identity element must work for all x.
I have to determine whether or not the Reals has an identity element with respect to *, given that x * y = |x - y|.
To do this, we were taught to solve x * e = x for e.
So x * e = |x - e| = x, which implies that x - e = x OR x - e = -x.
Thus e = 0 OR e = 2x.
Now we want to check that e * x = x * e = x.
e=0:
e*x = 0*x = |0-x| = |-x| = x
x*e = x*0 = |x-0| = x
e=2x:
e*x = |2x-x| = x
x*e = |x -2x| = |-x| = x.
They both work, but it doesn't make sense that there are two identity elements. What am I missing? Thanks for any help...