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I can't understand the difference between the definitions of Identity and Invertibility - can someone explain? I.e.: Identity For any element a in G there is

I can't understand the difference between the definitions of Identity and Invertibility - can someone explain? I.e.: Identity For any element a in G there is an element e where e⋆a = a⋆e = a. Invertibility For each element a in G there is an element b where a⋆b = b⋆a = e, where e is the identity.

SatvikBeri 10y ago
The integers >= 0 have 0 as the identity: 0 + n = n + 0 = n. But they don't have inverses, because e.g. there's no positive number n such that n + 1 = 0.
- cantankerous 10y ago
  I think you mean the naturals.
  - floatboth 10y ago
    People disagree about whether 0 is natural though. Better not to use the word at all :D
  - shele 10y ago
    OP said "integers >= 0". Not much to "think" here.
  - SatvikBeri 10y ago
    I wanted to avoid an argument about whether 0 is a natural number or not.
mdpopescu 10y ago
It's about where "e", the identity element, is placed - in the "identity" rule it is one of the terms of the operation (e * a, a * e) while in the "invertibility" rule it's the result of the operation. If you want an example, think Z (integers) and "+": there is an element of Z, namely zero (0), where the identity rule tells us that x + 0 = 0 + x = x and the invertibility rule says that for any x in Z, there is an y in z such that x + y = 0
lubomir 10y ago
Identity: there exists one unique element e such that for any a it holds that e⋆a = a⋆e = a. This e is the same no matter what a you pick. Invertibility: given any a, there is b such that a⋆b = b⋆a = e. For two different a's you get two different b's. As example let G be a set of all strings and the operation is appending. There is element e == "". For any string a it holds that if you append empty string to any side, you get the original string back. However there is no inverse. You can not append two string to form an empty string. Apparently, strings with appending do not form a group. Group example: let G be a set of all integers and the operation is addition. Then e is 0, as for all a it holds that a + 0 = 0 + a = a, and for each a you can find -a such that a + (-a) = (-a) + a = 0.
eklavya 10y ago
Think of identity as neutral (not good, not bad). Add neutral to good or bad and they remain the same. Invertibility is existence of Yin for a Yang, superhero for a supervillain, north pole for a south and combining them creates a neutral.

SatvikBeri 10y ago

The integers >= 0 have 0 as the identity: 0 + n = n + 0 = n. But they don't have inverses, because e.g. there's no positive number n such that n + 1 = 0.

cantankerous 10y ago
I think you mean the naturals.
- floatboth 10y ago
  People disagree about whether 0 is natural though. Better not to use the word at all :D
- shele 10y ago
  OP said "integers >= 0". Not much to "think" here.
- SatvikBeri 10y ago
  I wanted to avoid an argument about whether 0 is a natural number or not.

mdpopescu 10y ago

It's about where "e", the identity element, is placed - in the "identity" rule it is one of the terms of the operation (e * a, a * e) while in the "invertibility" rule it's the result of the operation. If you want an example, think Z (integers) and "+": there is an element of Z, namely zero (0), where the identity rule tells us that x + 0 = 0 + x = x and the invertibility rule says that for any x in Z, there is an y in z such that x + y = 0

lubomir 10y ago

Identity: there exists one unique element e such that for any a it holds that e⋆a = a⋆e = a. This e is the same no matter what a you pick. Invertibility: given any a, there is b such that a⋆b = b⋆a = e. For two different a's you get two different b's. As example let G be a set of all strings and the operation is appending. There is element e == "". For any string a it holds that if you append empty string to any side, you get the original string back. However there is no inverse. You can not append two string to form an empty string. Apparently, strings with appending do not form a group. Group example: let G be a set of all integers and the operation is addition. Then e is 0, as for all a it holds that a + 0 = 0 + a = a, and for each a you can find -a such that a + (-a) = (-a) + a = 0.

eklavya 10y ago

Think of identity as neutral (not good, not bad). Add neutral to good or bad and they remain the same. Invertibility is existence of Yin for a Yang, superhero for a supervillain, north pole for a south and combining them creates a neutral.