GATE CS 2025 Set 1 Q.49: Monoids and Groups

$A = \{0, 1, 2, 3, \dots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in F$ , we define

(f_1 \odot 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 $(F, \odot)$ ?

$(F, \odot)$ is an Abelian group.

$(F, \odot)$ is an Abelian monoid.

$(F, \odot)$ is a non-Abelian group.

$(F, \odot)$ is a non-Abelian monoid.

Official answer and solution

Official answer: B
From: Official GATE 2025 answer key

Solution

Check closure, associativity, identity element, inverses, and commutativity of pointwise addition on $A = \mathbb{N}_0$ .

Checking the algebraic properties of $(F, \odot)$ where $A = \{0, 1, 2, \dots\}$ :

Closure: For any $f_1, f_2 \in F$ and $n \in A$ , $f_1(n) + f_2(n) \in A$ , so $f_1 \odot f_2 \in F$ .
Associativity & Commutativity: Inherited directly from addition of non-negative integers.
Identity: The constant zero function $e(n) = 0$ belongs to $F$ and satisfies $(f \odot e)(n) = (e \odot f)(n) = f(n)$ .
Inverses: A function $f \in F$ with $f(n) > 0$ has no inverse in $F$ because $-f(n) < 0 \notin A$ .

Therefore, $(F, \odot)$ is an Abelian monoid (and not a group).

Final answer

Option (B)

Common trap

Assuming additive inverses exist without noticing that $A$ contains only non-negative integers.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 49 of 65
Section: Computer Science and Information Technology

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