Is this correct, or where did I go wrong in my answer.
Give a reason to support: The natural numbers {1,2,3,4,5,...}are not a group under addition.
My reason:
0 is not included in the group since it is not included by identity property it can not be conducted making this group of numbers not a group under addition.
http://en.wikipedia.org/wiki/Natural_number
set theorists disagree with you.
Aren't natural numbers also known as counting numbers?
Your reasoning is partially correct, but let me explain it in a more detailed way:
To determine whether the set of natural numbers {1, 2, 3, 4, 5, ...} forms a group under addition, we need to consider the four group axioms: closure, associativity, identity, and inverse.
1. Closure: For a set to be a group under a certain operation, the result of that operation on any two elements in the set must also be in the set. In this case, the operation is addition. For example, 1 + 2 = 3, which is indeed a natural number. So, in terms of closure, the natural numbers form a group as long as the sum of any two natural numbers is also a natural number.
2. Associativity: The addition operation is associative, meaning that for any three elements a, b, and c in the set, (a + b) + c = a + (b + c). Since addition is associative for natural numbers, this criterion is met.
3. Identity: A group must have an identity element, which is an element that, when combined with any other element in the group, leaves that other element unchanged. In the case of addition, the identity element is 0. However, in your answer, you claim that 0 is not included in the group. This is incorrect. In fact, 0 is a natural number, and since adding 0 to any other natural number leaves it unchanged, 0 is the identity element for the natural numbers under addition.
4. Inverse: For every element in a group, there must exist an inverse element that, when combined with the original element, gives the identity element. In this case, since we are considering addition, the inverse of any natural number n is -n. For example, the inverse of 3 is -3, because 3 + (-3) = 0, the identity element. Therefore, the natural numbers do have inverses under addition.
In summary, the natural numbers {1, 2, 3, 4, 5, ...} do form a group under addition because they satisfy the closure, associativity, identity, and inverse properties. Contrary to your reasoning, the inclusion of 0 as a natural number does not hinder it from being a group under addition.