To determine if the set of positive integers (denoted as ℕ+ ) forms a group under the operation of addition, we need to evaluate it against the four fundamental properties that define a mathematical group:
- Closure: A set is closed under an operation if performing that operation on members of the set results in another member of the same set. For positive integers, if you add two positive integers together (e.g., 3 + 5 = 8), the result (8) is also a positive integer. Therefore, the set of positive integers is closed under addition.
- Associativity: An operation is associative if changing the grouping of the numbers does not change the result. In simpler terms, for any three positive integers a, b, and c, it must hold that (a + b) + c = a + (b + c). Since addition is universally recognized as an associative operation, this property holds true for positive integers.
- Identity Element: A group requires the existence of an identity element, a member of the set that, when combined with any member of the set in the operation, yields that member itself. The identity element for addition is 0 (because a + 0 = a). However, 0 is not a member of the set of positive integers. Therefore, this property fails for the set of positive integers.
- Inverse Element: For each member of the set, there must be an inverse element such that when the two are combined using the operation, the result is the identity element. In the case of addition, if we take a positive integer a, we need to find an element b such that a + b = 0. The only number that satisfies this is -a, which is not a positive integer. Thus, the set of positive integers also fails to meet this condition.
In conclusion, while the set of positive integers is closed under addition and the operation is associative, it does not have an identity element (0 is not included) nor inverse elements (no positive integer combines to yield 0). Therefore, the set of positive integers does not qualify as a group under the operation of addition.