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corey1234
 2 years ago
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(xy). Now when I do it, I break up the absolute value into xe for xe greater than 0 and (xe) for xe less than 0, where e is the identity element. Please explain your steps to this problem.
corey1234
 2 years ago
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(xy). Now when I do it, I break up the absolute value into xe for xe greater than 0 and (xe) for xe less than 0, where e is the identity element. Please explain your steps to this problem.

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ryan16
 2 years ago
Best ResponseYou've already chosen the best response.0The identity element is 0

corey1234
 2 years ago
Best ResponseYou've already chosen the best response.0No. Only if you restrict the domain to positive real numbers.

Stiwan
 2 years ago
Best ResponseYou've already chosen the best response.0I dont get why you're not satisfied with 0. The absolute value of x0 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.

ryan16
 2 years ago
Best ResponseYou've already chosen the best response.0This operation has identity zero if the domain is nonnegative R, and no identity in general if domain is all R. Because abs(x0)=abs(0x)=x (if x<0) which doesnt equal x

corey1234
 2 years ago
Best ResponseYou've already chosen the best response.0Yes. Ryan got it right. That is correct.

jickso
 2 years ago
Best ResponseYou've already chosen the best response.0Well, 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: xe=x > e=0 2) x=0: 0e=0 > e=0 3) x<0: xe=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.
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