A = {0, 1, 2, 3, …} is the set of non-negative integers. Let F be the set of functions from A to its

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Question

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, (f₁ Θ f₂)(n) = f₁(n) + f₂(n) for every n ∈ A. Which of the following is/are CORRECT about (F, Θ)?

Accepted Answer

Correct Answer: (B) (F, Θ) is an Abelian monoid.Concept — Algebraic Structures:Semigroup: Closed set with an associative binary operation.Monoid: Semigroup with an identity element.Group: Monoid where every element has an inverse.Abelian (commutative): The operation satisfies a⊕b = b⊕a for all a, b.Verifying properties of (F, Θ):1. Closure: For f₁, f₂ ∈ F, (f₁ Θ f₂)(n) = f₁(n) + f₂(n). Since f₁(n) and f₂(n) are non-negative integers, their sum is also a non-negative integer. Therefore f₁ Θ f₂ maps A to A → F is closed under Θ. ✓2. Associativity: ((f₁ Θ f₂) Θ f₃)(n) = (f₁(n)+f₂(n))+f₃(n) = f₁(n)+(f₂(n)+f₃(n)) = (f₁ Θ (f₂ Θ f₃))(n). Addition of integers is associative → Θ is associative. ✓3. Identity element: Define z: A → A by z(n) = 0 for all n ∈ A. Then (f Θ z)(n) = f(n) + 0 = f(n) and (z Θ f)(n) = 0 + f(n) = f(n). So z is the identity element → identity exists. ✓→ (F, Θ) is a Monoid.4. Commutativity: (f₁ Θ f₂)(n) = f₁(n)+f₂(n) = f₂(n)+f₁(n) = (f₂ Θ f₁)(n). → Θ is commutative. ✓→ (F, Θ) is an Abelian Monoid.5. Inverses (does it form a Group?): For a function f to have an inverse g under Θ, we need f(n) + g(n) = 0 for all n. This requires g(n) = −f(n) for all n. However, since the codomain is A = {0,1,2,...} (non-negative integers), g(n) = −f(n) is negative when f(n) > 0 — which is outside A. Therefore, most functions in F do NOT have inverses → (F, Θ) is NOT a group.Conclusion: (F, Θ) is an Abelian Monoid (Option B only).

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, (f₁ Θ f₂)(n) = f₁(n) + f₂(n) for every n ∈ A. Which of the following is/are CORRECT about (F, Θ)?

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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, (f₁ Θ f₂)(n) = f₁(n) + f₂(n) for every n ∈ A. Which of the following is/are CORRECT about (F, Θ)?

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