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.

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- jim_thompson5910

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.

- jim_thompson5910

I guess I should say though that if you're going for one unique identity element, then it would simply be 0

- anonymous

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.

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## More answers

- jim_thompson5910

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

- jim_thompson5910

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

- jim_thompson5910

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

- anonymous

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 x0. 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.

- jim_thompson5910

yes, but we found that e = 2x
and I showed that x = 0
so naturally e = 2x ---> e = 2*0 ----> e = 0

- jim_thompson5910

x * 0 = x for all x (when x > 0 or x < 0)

- anonymous

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 xe, 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.

- jim_thompson5910

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|

- anonymous

Ok so what you are saying is that e*x== e-x when xe and since e==0 and x>0 , then 0*x== x-0 which is just x?

- jim_thompson5910

yes, sry i was trying a different way to do this, but not really finding anything

- anonymous

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?

- jim_thompson5910

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

- anonymous

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.

- jim_thompson5910

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

- anonymous

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.

- jim_thompson5910

you want to find an identity element (e) for x * y = |x - y|
not for x * y = x - y

- jim_thompson5910

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

- jim_thompson5910

then we check to see if e = 0 is actually the true identity element for x * y = |x - y|

- anonymous

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?

- jim_thompson5910

yes x > 0 means |x - 0| = x - 0, but we want something that will apply to both x > 0 and x < 0

- anonymous

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.

- jim_thompson5910

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

- anonymous

So then there is no identity element for the set of real numbers for this operation.?

- jim_thompson5910

yes it looks like it since it doesn't apply to all real numbers

- jim_thompson5910

personally, I would restrict the domain and say e = 0 is the identity element, but it looks like there isn't one here

- anonymous

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?

- jim_thompson5910

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

- jim_thompson5910

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

- anonymous

But what about x-e==x. That can have negative x values. And x>0. Is that what you are saying?

- jim_thompson5910

yes x - e = x can have negative values, but we're referring to |x-y| not x-y

- jim_thompson5910

this is why it's very important to check the possible solutions

- anonymous

Ok thank you so much.

- jim_thompson5910

np

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