GATE CS 2025 Set 1 — Question 49
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Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields
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Question
is the set of non-negative integers. Let be the set of functions from to itself. For any two functions, , we definefor every number in . Which of the following is/are CORRECT about the mathematical structure ?
Correct answer
(B) (F,) is an Abelian monoid.
Solution
Let's evaluate the properties of the structure :
1.Closure: For any , their pointwise sum is also a function from to because the sum of two non-negative integers is a non-negative integer. Closure holds.
2.Associativity: Pointwise addition is associative because addition in is associative.
3.Identity: The zero function for all acts as the identity. Since , .
4.Commutativity: Pointwise addition is commutative because addition in is commutative. Thus, the structure is Abelian.
5.Inverse: For a function to have an inverse , we need for all , which implies . However, if , then , which is not in the codomain . Thus, inverses do not exist for all elements.
Since it has identity and associativity but lacks inverses, it is a monoid, not a group. Being commutative, it is an Abelian monoid.Continue learning with Success Tracker
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