Counterexample to uniqueness of identity element?

In summary, the conversation discusses the uniqueness of the identity element in a binary operation and how it relates to the concept of a largest or smallest element in a set. It is clarified that individual elements in a set do not have their own identities and that an identity must satisfy specific conditions for all elements in the set. The example of the max operation is used to illustrate this concept.
  • #1
math771
204
0
(Hopefully, this question falls under analysis. I was unable to match it well with any of the forums.)
The proof that the identity element of a binary operation, f: X x X [itex]\rightarrow[/itex] X, is unique is simple and quite convincing: for any e and e' belonging to X, e=f(e,e')=f(e',e)=e'.
However, if we take f(m,n)=max(m,n), it appears that any m will have multiple identity elements--the elements of the set of numbers n less than m.
There must be something that I'm missing here. Any help would be appreciated. Thanks!
 
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  • #2
You are missing the fact that individual elements of the set do not "have" their own identities. In order that "e" be the identity, we must have f(e, a)= f(a, e) for all a in the set. f(a,b)= max(a, b), over some ordered set, has an identity if and only if the set has an largest member.
 
  • #3
Thank you. :smile:
 
  • #4
HallsofIvy said:
You are missing the fact that individual elements of the set do not "have" their own identities. In order that "e" be the identity, we must have f(e, a)= f(a, e) for all a in the set. f(a,b)= max(a, b), over some ordered set, has an identity if and only if the set has an largest member.

Wouldn't the "max identity" be the smallest element? For example in the natural numbers 0, 1, 2, ..., we have max(0, n) = n for all n, so 0 is the identity of the max operation.
 
  • #5
I was thinking the same thing. Maybe that's what HallsofIvy meant.
 
  • #6
SteveL27 said:
Wouldn't the "max identity" be the smallest element? For example in the natural numbers 0, 1, 2, ..., we have max(0, n) = n for all n, so 0 is the identity of the max operation.

Correct. Assuming the smallest element is e, then max(e,a) = max(a,e) = a for all a in the set. If you're dealing with min, then you do need a largest element in order to have an identity that works for all numbers.
 
  • #7
Yes, of course. I don't know why I said "maximum".
 

1. What is a counterexample to uniqueness of identity element?

A counterexample to uniqueness of identity element is a mathematical construct that disproves the notion that there can only exist one identity element in a set with a binary operation. It shows that there can be multiple elements that satisfy the definition of an identity element.

2. How does a counterexample to uniqueness of identity element challenge the concept of identity element?

It challenges the concept of identity element by demonstrating that the existence of multiple elements that satisfy the definition of an identity element goes against the traditional understanding of identity element as a unique and singular entity. This highlights the importance of carefully defining and understanding mathematical concepts and the potential for exceptions to common assumptions.

3. Can you provide an example of a counterexample to uniqueness of identity element?

Yes, consider the set of even integers with the operation of multiplication. While the number 1 is traditionally thought of as the identity element for multiplication, in this set, both 1 and -1 satisfy the definition of an identity element, making it a counterexample to uniqueness of identity element.

4. How does the concept of a counterexample to uniqueness of identity element impact other mathematical concepts?

The concept of a counterexample to uniqueness of identity element challenges traditional assumptions and can lead to a deeper understanding of other mathematical concepts. It can also prompt the development of new ideas and theories, further expanding the breadth and depth of mathematics.

5. Why is it important for scientists to be aware of counterexamples to uniqueness of identity element?

It is important for scientists to be aware of counterexamples to uniqueness of identity element because it highlights the importance of rigorous definitions and critical thinking in mathematics. It also serves as a reminder that no concept should be taken for granted and that there is always room for further exploration and discovery.

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