More than one identity element for absolute value?

[physicsforum7]

I was thinking about identities, and seem to have arrived at a contradiction. I'm sure I'm missing something.

A(n) (two-sided) identity for a binary operation must be unique.

I will reproduce the familiar proof:

Proof: Suppose a is an arbitrary element of a set S, e and e' are both identities, and * is an arbitrary binary operation. Then a*e=e'*a=a. Now take a=e' in the first equation; so e'*e=e'. Take a=e in the second equation; e'*e=e. Thus e'*e=e=e'.

But what about x*y= $\left|x-y\right|$ defined on the set of nonnegative real numbers?

It seems that both 0 and 2x are identities.

Can anybody find my mistake? Thanks ahead of time.

 

[micromass]

I was thinking about identities, and seem to have arrived at a contradiction. I'm sure I'm missing something.

A(n) (two-sided) identity for a binary operation must be unique.

I will reproduce the familiar proof:

Proof: Suppose a is an arbitrary element of a set S, e and e' are both identities, and * is an arbitrary binary operation. Then a*e=e'*a=a. Now take a=e' in the first equation; so e'*e=e'. Take a=e in the second equation; e'*e=e. Thus e'*e=e=e'.

But what about x*y= $\left|x-y\right|$ defined on the set of nonnegative real numbers?

It seems that both 0 and 2x are identities.

Can anybody find my mistake? Thanks ahead of time.

You're right that 0 is an identity. But 2x is not an identity. Indeed, 2x isn't even a real number! Rather, x is a variable, so 2x is a variable.

Think about what number you actually mean with 2x. What is it's decimal representation? Is it bigger than 1?? All these questions should convince you that 2x makes no sense as a number.

 

[physicsforum7]

You're right that 0 is an identity. But 2x is not an identity. Indeed, 2x isn't even a real number! Rather, x is a variable, so 2x is a variable.

Think about what number you actually mean with 2x. What is it's decimal representation? Is it bigger than 1?? All these questions should convince you that 2x makes no sense as a number.

Aha! But I'm not entirely convinced yet. What of inverses? Isn't -x the inverse of x with respect to addition in the set of real numbers?

Surely 2x is as much a number as -x... right?

 

[micromass]

Aha! But I'm not entirely convinced yet. What of inverses? Isn't -x the inverse of x with respect to addition in the set of real numbers?

Surely 2x is as much a number as -x... right?

But -x is not a number either!

If we say that -x is the inverse of x, then this is shorthand for: given a real number denoted by x, then -x is the inverse of x.
So in this sense, x just becomes the name of a real number. For example: given the real number 2, then -2 is the inverse of 2. Or given the real number 4, then -4 is the inverse of 4. So the convention is that x can take on every real number.

When you are saying that 2x is the identity of *. What is it that you mean?? What is x? Can we let x take on every real number?? The notation does not make any sense.

 

[physicsforum7]

But -x is not a number either!

If we say that -x is the inverse of x, then this is shorthand for: given a real number denoted by x, then -x is the inverse of x.
So in this sense, x just becomes the name of a real number. For example: given the real number 2, then -2 is the inverse of 2. Or given the real number 4, then -4 is the inverse of 4. So the convention is that x can take on every real number.

When you are saying that 2x is the identity of *. What is it that you mean?? What is x? Can we let x take on every real number?? The notation does not make any sense.

I hope you'll pardon my slowness; I still don't understand the second paragraph you wrote. When you say,

"When you are saying that 2x is the identity of *. What is it that you mean?? What is x? Can we let x take on every real number?? The notation does not make any sense."

I am confused, because I said in my first post that x can take on the value of any nonnegative real number.

"But what about x*y= |x−y| defined on the set of nonnegative real numbers?

It seems that both 0 and 2x are identities."

Perhaps I could have been more explicit, but that is what I meant.

 

[micromass]

I hope you'll pardon my slowness; I still don't understand the second paragraph you wrote. When you say,

"When you are saying that 2x is the identity of *. What is it that you mean?? What is x? Can we let x take on every real number?? The notation does not make any sense."

I am confused, because I said in my first post that x can take on the value of any nonnegative real number.

"But what about x*y= |x−y| defined on the set of nonnegative real numbers?

It seems that both 0 and 2x are identities."

Perhaps I could have been more explicit, but that is what I meant.

OK, so x can be 1?? So you say that 2 is an identity?? This is not true as 2*0≠0.

 

[physicsforum7]

OK, so x can be 1?? So you say that 2 is an identity?? This is not true as 2*0≠0.

No, that's not what I'm saying. Rather, given the nonnegative real number 2, 4 is an identity of 2. Because 2*4=2=4*2.

But I think I've just discovered my mistake. Let me know if this is right: an identity e is a SINGLE element of a set S for which a*e=e*a=a FOR EVERY a in S. In other words, an identity - in order to be such - must be

(A) a single element of S and

(B) hold for, not just one, but EVERY element in S.

On the other hand, an inverse is specific - though it need not be unique to a single element of S - to a single element in S. In other words,

(1) While each element a in S has only one inverse, an element that acts as an inverse for a may be the same element that acts as an inverse for b in S.

(2) From the theorem of uniqueness: an element a may only have one element e as its inverse.

Now I'm not wholly sure of (1), for there may be a theorem that I've yet to encounter. (If so, can you correct me?) But strictly from the definitions, would you say that this is all accurate?

If so, I think this clears up my confusion.

 

[micromass]

No, that's not what I'm saying. Rather, given the nonnegative real number 2, 4 is an identity of 2. Because 2*4=2=4*2.

But I think I've just discovered my mistake. Let me know if this is right: an identity e is a SINGLE element of a set S for which a*e=e*a=a FOR EVERY a in S. In other words, an identity - in order to be such - must be

(A) a single element of S and

(B) hold for, not just one, but EVERY element in S.

YES! That's exactly it! An identity is ONE elements that is good for EACH other element.

On the other hand, an inverse is specific - though it need not be unique to a single element of S - to a single element in S.

Right. Here we have that EACH element has a specific inverse for THAT element.

In other words,

(1) While each element a in S has only one inverse, an element that acts as an inverse for a may be the same element that acts as an inverse for b in S.

Right. Nothing in the definitions rules out that the inverse of a cannot be equal to the inverse of b.
However, this can actually not happen (but it is not straightforward from the definitions, we need to prove it). We have that: if the inverse of a equals the inverse of b, then a=b and the two inverses are equal.
The proof of this fact follows from the fact that the inverse of the inverse of a equals a.
If I denote the inverse of a by -a. Then we have that -(-a)=a.
So, if -a=-b. Then -(-a)=-(-b) and thus a=b.

(2) From the theorem of uniqueness: an element a may only have one element e as its inverse.

Right. But this is again not clear from the definitions. It is again a theorem that must be proven.

 

[physicsforum7]

Excellent! Thank you very much for your patience and your help.