\(𝐴 = \{0, 1, 2, 3, … \}\) is the set of non-negative integers. Let \(F\) be…
2025
\(𝐴 = \{0, 1, 2, 3, … \}\) is the set of non-negative integers. Let \(F\) be the set of functions from \(A\) to itself. For any two functions, \(𝑓_1, 𝑓_2 ∈ Ϝ\), we define
\((𝑓_1⨀𝑓_2)(𝑛) = 𝑓_1(𝑛) + 𝑓_2(𝑛)\)
for every number \(n\) in \(A\). Which of the following is/are CORRECT about the mathematical structure \((Ϝ, ⨀)\)?
- A.
\((Ϝ, ⨀)\)is an Abelian group. - B.
\((Ϝ, ⨀)\)is an Abelian monoid. - C.
\((Ϝ, ⨀)\)is a non-Abelian group. - D.
\((Ϝ, ⨀)\)is a non-Abelian monoid.
Attempted by 86 students.
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Correct answer: B
Key facts: The operation is pointwise addition: (f1 ⨀ f2)(n) = f1(n) + f2(n) for each n in A (the set of non-negative integers).
Closure: For any two functions f1,f2 in F, f1(n)+f2(n) is a non-negative integer for every n, so f1 ⨀ f2 is in F.
Associativity: Integer addition is associative, so pointwise addition of functions is associative.
Identity element: The zero function 0 defined by 0(n)=0 for all n is the identity because f ⨀ 0 = f for every f in F.
Commutativity: Integer addition is commutative, hence pointwise addition is commutative.
Lack of inverses in general: For a function f with f(n)>0 for some n, there is no g in F with f(n)+g(n)=0 because g(n) would have to be negative. Example: the constant-1 function has no inverse in F.
Conclusion: The structure (F, ⨀) satisfies closure, associativity, identity, and commutativity, so it is an Abelian (commutative) monoid. It is not a group because most elements do not have additive inverses in the set of non-negative integers.
Assessment of the given statements:
The statement that the structure is an Abelian monoid is correct.
The statements that the structure is an Abelian group, a non-Abelian group, or a non-Abelian monoid are incorrect (it is not a group, and it is commutative, so it cannot be non-Abelian).