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GATE 2025 CS (CS1) – Question 49

Engineering Mathematics · Discrete Mathematics: Monoids and Groups · 2 marks · Multiple select

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

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

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Show answer and explanation

Correct answer: (B) (F, $\odot$) is an Abelian monoid.

Explanation

Pointwise addition is closed, associative and commutative, and the zero function is the identity. Inverses would need negative values, which are not in A, so it is not a group. It is an Abelian monoid.