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x * y = |x - y| x * e = |x - e| x = |x - e| ------------------------------------------------------- If x - e > 0, then x = x - e x-x = -e 0x = -e 0 = -e -e = 0 e = 0 ------------------------------------------------------- If x - e < 0, then x = -(x - e) -x = x - e -x-x = -e -2x = -e -e = -2x e = 2x So the identity element is either e = 0 or e = 2x for any real x.
I guess I should say though that if you're going for one unique identity element, then it would simply be 0
Yes, but the definition of an identity element for R is that x*e==e*x==x for ALL of R. The e=0 is only true for x greater than 0 and the e=2x is only true for x greater than 0 too. That is the confusing part.
2x is only an identity for x, so 2x doesn't work for all x 0 works for all x so 0 is an identity of x*y
i should clarify, e = 2x only satisfies x * e = 0 for any x since it doesn't work for e * x = 0, it starts to fall apart
if you repeat the steps above, but do it for e * x instead of x * e, you'll find that e = 2x when e - x > 0 so e = 0 when x - e > 0 and e = 2x when e - x > 0 this means 2x = 0 ----> x = 0
I am not so sure what you are talking about. You are not trying to solve for x. You are trying to solve for e. Now for x>e then e==0 and for x
0. aND X*0==X AND X* 2X ==-X. So e=2x is automatically out of the question. But for e=0, x*0 only equals x when x>0.
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)
Ok I am kind of confused now. Can we start this from the beginning. So we are looking to see if there is an identity element for x*y==|x-y|. We check |x-e|, which equals x-e when x>e and -(x-e) when x
e, then x-e==x and e==0. So that means if x>0, then x*0==x and 0*x==-x. This is a problem. e=0 only works as an identity element when x*0 not 0*x.
Now we check for x< e. Then we agree that e==2x. So x*2x== -x for x>0 and x for x<0. So again, it is only x for when x is less than 0. So this is not an identity element either.
when you say "e=0 only works as an identity element when x*0 not 0*x. " that's just not true IF x * y = x - y, then it would be true BUT the binary operation is x * y = |x - y| so x * 0 = |x - 0| x * 0 = |x| x * 0 = x and 0 * x = |0 - x| 0 * x = |-x| 0 * x = x So 0 is clearly an identity element for x * y = |x - y|
Ok so what you are saying is that e*x== e-x when x
e and since e==0 and x>0 , then 0*x== x-0 which is just x?
yes, sry i was trying a different way to do this, but not really finding anything
Also, when we first said that x>0, does that mean that that condition has to be true in order for x-0=x? Because x can be negative there?
yes you have to force x to be positive because this example below shows where things break down 0 * x = |0 - x| say x = -2 0 * (-2) = |0 - (-2)| 0 * (-2) = |0 + 2| -2 = 2
yes but then x*e==(x-e)=x works when x is negative if e==0. For example x-0=x, x can be negative there.
you have to go back to the definition of what the binary operator is everytime it works for x - 0 = x but it doesn't work for |x - 0| = x
But why do you have to go back to the definition everytime. I mean, you used the fact that |x-e|==(x-e)=x to solve for e. And you found out that e=0. So shouldn't that condition hold for x-0=x? But it doesn't since x can be negative there.
you want to find an identity element (e) for x * y = |x - y| not for x * y = x - y
we break down |x-y| into its smaller parts to find the various possible solutions, then we piece them back together to find that only e = 0 is the identity element
then we check to see if e = 0 is actually the true identity element for x * y = |x - y|
So what you are saying is that x>0 is only a condition for |x-e| to equal (x-e), and the fact that x-e=x is something totally different to just find e, which equals 0?
yes x > 0 means |x - 0| = x - 0, but we want something that will apply to both x > 0 and x < 0
But all of this works only when x>0. Which means e=0 is not an identity element because it is not true for all x in R.
good point, if you restrict the domain to the set of positive numbers, then it works if the domain is the set of all real numbers, then it won't work because of the negative numbers
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
personally, I would restrict the domain and say e = 0 is the identity element, but it looks like there isn't one here
It just doesn't make sense to me how x>0 for |x-e|=x-e but x can be nagative for x-e=x. Then there is a contradiction right?
yeah in one sense, we're saying e = 0 is an identity element (which is supposed to be unique), but in another, we're saying e = 2x which says that we have infinitely many identity elements
one quick way to determine there is no identity is this x * e = |x - e| x = |x - e| which only has solutions when x > 0 like you said (since k = |x| only has solutions when k > 0) but x is allowed to be negative, which contradicts this fact so there are no identity elements
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.