The set of integers is an abelian group under?
- Addition
- Multiplication
- Division
- None
Explanation
The set of integers, denoted by Z, forms an abelian group under the operation of addition (+). This means that the following properties hold:
- Closure: For any integers a, b, their sum a + b is also an integer.
- Associativity: (a + b) + c = a + (b + c) for any integers a, b, c.
- Commutativity: a + b = b + a for any integers a, b.
- Identity: There exists an identity element, which is 0, such that a + 0 = a for any integer a.
- Inverse: For each integer a, there exists an inverse element, which is -a, such that a + (-a) = 0.
The set of integers does not form a group under multiplication, because the inverse property does not hold (except for 1 and -1). Division is not even a binary operation on the set of integers, because it is not defined for all pairs of integers (e.g., 1/2 is not an integer).
Therefore, the correct answer is Addition.
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