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A = {0, 1, 2, 3,...} is the set of non-negative integers. Let F be the set of functions from A to itself. For any two functions, f1, f2 ∈ F, we define (f1 ⊙ f2)(n) = f1(n) + f2(n) for every number n in A. Which of the following is/are CORRECT about the mathematical structure (F, ⊙)?

1. (F, ⊙) is an Abelian group.
2. (F, ⊙) is an Abelian monoid.
3. (F, ⊙) is a non-Abelian group.
4. (F, ⊙) is a non-Abelian monoid.

Correct answer: (F, ⊙) is an Abelian monoid.

Solution

(F, ⊙) is an Abelian monoid because it satisfies the properties of closure, associativity, and the existence of an identity element (the zero function), and the operation ⊙ is commutative, meaning the order of functions does not affect the result.

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