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.
MIT 18.06 Linear Algebra, Spring 2010
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The identity element is 0
No. Only if you restrict the domain to positive real numbers.
I dont get why you're not satisfied with 0. The absolute value of |x-0| as well as the absolute value of |x+0| is |x|. It doesn't matter whether x is positive or negative, this always holds. Hence x*e = e*x = x for e = 0, and 0 clearly is the identity element.
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This operation has identity zero if the domain is nonnegative R, and no identity in general if domain is all R. Because abs(x-0)=abs(0-x)=-x (if x<0) which doesnt equal x
Yes. Ryan got it right. That is correct.
Well, it's much easier to see that with a concrete counterexample, am I not right? Let's see, now, exempli gratia, take x=-5. Let us suppose that e=0. Then, x * e = -5 * 0 = | -5 - 0 | = |-5| = 5 =/= -5.
To make things more clear, logicaly... You can break down absolute value into three parts, one for x>0, second for x=0 and third for x<0 (of course, we observe R as the set on which the binary operation is defined). This is a disjunction, therefore if we were to conclude anything from it by deduction, we must obtain the same result for every branch, yet we do not:
1) x>0: x-e=x -> e=0
2) x=0: 0-e=0 -> e=0
3) x<0: x-e=-x -> e=2x
Now, we do not get the -same- conclusion for every member of disjunction, therefore we can only conclude that it is either e=0 or e=2x. But can we talk about e=2x at all? No, because it is not the condition for a neutral element that "for every x exists e" (in this case we would also be able to restrict it to x<0), but that "exists e such that for every x", and there is a big difference. Note that the condition for the inverse elements is similar to the first case "for every x there exists -x"...
Conclusion, neutral element would exist on the set of nonnegative real numbers.