GATE2025_CS1_Q49

A = {0, 1, 2, 3, ... } is the set of non-negative integers. Let F be the set of all functions from A to itself. For any two functions, f₁, f₂ ∈ F, we define

(f₁ ᴏ f₂)(n) = f₁(n) + f₂(n)

for every number n in A. Which of the following is/are CORRECT about the mathematical structure (F, ᴏ)?

Note: This is a Multiple Select Question (MSQ). One or more options may be correct.

✅ Final Answer:

The correct option is (B).

🔹 Short Answer:

The structure (F, ᴏ) is checked for standard algebraic properties. It satisfies closure, associativity, identity (the zero function), and commutativity. It is therefore an Abelian monoid. It fails to be a group because for any function `f` that produces a non-zero output, its additive inverse `-f` is not in F, as its output would not be a non-negative integer.

🔸 Long Answer:

To determine the nature of the mathematical structure (F, ᴏ), we need to test it against the properties of groups and monoids (Closure, Associativity, Identity, and Inverse) and check for commutativity (the Abelian property).

Property	Status	Reasoning
Closure	✔ Satisfied	If f₁ and f₂ map non-negative integers (A) to non-negative integers (A), their sum f₁(n) + f₂(n) will also be a non-negative integer. So, (f₁ ᴏ f₂) is also a function in F.
Associativity	✔ Satisfied	The operation is based on integer addition, which is associative. For any n, ((f₁ᴏf₂)(n) + f₃(n)) = (f₁(n)+f₂(n))+f₃(n) = f₁(n)+(f₂(n)+f₃(n)) = (f₁ᴏ(f₂ᴏf₃))(n).
Identity Element	✔ Satisfied	There exists a zero function, z(n) = 0 for all n ∈ A. This function is in F. For any function f ∈ F, (f ᴏ z)(n) = f(n) + z(n) = f(n) + 0 = f(n). So, the zero function is the identity element.
Inverse Element	❌ Not Satisfied	For a function f to have an inverse f⁻¹, we need (f ᴏ f⁻¹)(n) = 0 for all n. This means f(n) + f⁻¹(n) = 0, so f⁻¹(n) = -f(n). Consider a non-zero function like f(n) = 1. Its inverse would be f⁻¹(n) = -1. However, the output -1 is not in A, so the inverse function f⁻¹ is not in F. Thus, not every element has an inverse.
Commutativity (Abelian)	✔ Satisfied	The operation is based on integer addition, which is commutative. (f₁ ᴏ f₂)(n) = f₁(n) + f₂(n) = f₂(n) + f₁(n) = (f₂ ᴏ f₁)(n).

Conclusion

• A structure with Closure, Associativity, and Identity is a Monoid.
• Since the Inverse property fails, it is not a Group.
• Since it is Commutative, it is an Abelian Monoid.

Based on this analysis, we evaluate the options:

• (A) (F, ᴏ) is an Abelian group. – False, it lacks inverses.
• (B) (F, ᴏ) is an Abelian monoid. – True.
• (C) (F, ᴏ) is a non-Abelian group. – False, it's not a group and it is Abelian.
• (D) (F, ᴏ) is a non-Abelian monoid. – False, it is an Abelian monoid.

Even though this is an MSQ, only one statement correctly describes the structure.

Bibliography:
- Gallian, J. A. (2017). Contemporary Abstract Algebra. Cengage Learning. (Chapters on Groups and Rings).
- Rosen, K. H. (2018). Discrete Mathematics and Its Applications. McGraw-Hill Education. (Chapter 9: Relations).

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