Talk:Identity function

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 Field:  Algebra

I changed back to the previous version for the following reasons:

  • The identity function is in general not multiplicative. Only in the special case of M = positive integers could it be called a multiplicative function. But there are many other sets M out there.
  • We don't use colors in formulas
  • Blackboard bold is reserved for sets of numbers, like R or C. Letters that stand for functions, sets or variables are normally written in italic.

AxelBoldt, Thursday, April 18, 2002

I gave the notation 1M and a reference. Who uses the notation idM ? Multiplication by one is not restricted to positive integers but apply to any group. Bo Jacoby 10:43, 12 October 2005 (UTC)

I've rewritten the notation bit a little, and removed the reference to Jean-Marie Souriau. There is no need to mention a specific user of either notation, since both notations are common I think, for example: Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories uses idM, while Herrlich, Horst and Strecker, George E.; Category Theory, Allen and Bacon, Inc. Boston (1973), uses 1M. — Paul August 18:39, 13 October 2005 (UTC)
I've also restored the fact that:
The identity function on the positive integers is a multiplicative function (essentially multiplication by 1), considered in number theory.
I believe this statement is correct. I don't understand why it was removed. — Paul August 18:39, 13 October 2005 (UTC)

Your statement is just a very special case of the more general statement regarding vector spaces. That's why I removed it. There is no reason for restricting the integers to be positive. Nor is there a reason for restricting the numbers to be integers. Nor is there a reason for restricting the vectors to be numbers. In every case where multiplication by 1 makes sense, it represents an identity function. See my point ? I don't mind your removing my reference. (Someone might request a reference if I didn't provide it). Bo Jacoby 09:12, 14 October 2005 (UTC)

Yes in any algebraic structure which possesses a multiplicative identity, multiplication by that identity will be the identity function, but such functions are not generally called multiplicative. The reason for restricting to positive integers is because that is the ony context in which a multiplicative function is defined. The term is not, to my knowledge, used outside of number theory. Paul August 16:54, 14 October 2005 (UTC)

OK, now I see what you mean! I might not be the only reader who get more confused than enlightened by this reference to advanced number theory in an extremely elementary context. How many of your readers do you expect to look for this information under the heading Identity function ? I think none. Bo Jacoby 10:04, 17 October 2005 (UTC)

Merging with inclusion map[edit]

I disagree with a merge. Yes, these are related functions, actually both of them work by f(x)=x. However, the two articles look at the matter from a very different perspective. Typically one uses inclusion maps when one thinks of embedding a space into another, bigger space. The identity function on the other hand shows up when one deals with automorphisms of a given space, and related business. That is to say, it is true that both the identity function and the inclusion map have the formula f(x)=x, but that's all they have in common. Oleg Alexandrov (talk) 11:08, 20 October 2005 (UTC)

I agree with Oleg, I think these article should stay separate. Paul August 19:32, 20 October 2005 (UTC)
The identity map 1A and the inclusion map 1A is exactly the same thing. The articles should explain that to the reader. There is no mathematical reason to distinguish. There might be a historical reason, I don't know that. Bo Jacoby 13:06, 21 October 2005 (UTC)
The identity map is surjective i.e. onto, whereas the inclusion map 1A:A -> B is not, except for the trivial case where A = B. So they are not the same thing. Paul August 13:38, 21 October 2005 (UTC)

That is interesting. I leaned that function f equals function g if def(f)=def(g) and f(x)=g(x) for all x in def(f). f is injective if f(x)=f(y) implies x=y. f is surjective on B if for every y in B there exists an x in def(f) such that y=f(x). For example. f(x)=x2 (x in R), is not surjective on R, but is surjective on R+. So, strictly speaking, surjectivity is not a property of the function, but of the function f together with the codomain B. Is there any point in distinguishing functions having the same domain and the same values for the same arguments ? Bo Jacoby 17:37, 23 October 2005 (UTC)

Of course there is a point. Two functions are equal if they have the same domain, same codomain, and same output for given input. So, strictly speaking, you are incorrect; being surjective is part of what the function is about, not part of the codomain. Oleg Alexandrov (talk) 23:22, 23 October 2005 (UTC)

Why is it useful?[edit]

Could someone, please, explain, in plain English why the Identity function is useful?

What kind of applications does it have? — Preceding unsigned comment added by (talk) 01:58, 13 October 2012 (UTC)