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 f...

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 f_1, f_2 in F, we define (f_1 (-) 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, (-)) ? A. (F, (-)) is an Abelian group.  B. (F, (-)) is an Abelian monoid.  C. (F, (-)) is a non-Abelian group.  D. (F, (-)) is a non-Abelian monoid.

Accepted Answer

(F, (-)) is an Abelian monoid (Option B). The operation (-) adds two functions value-by-value: the new function's output at n is f_1(n) + f_2(n). Because ordinary integer addition is associative and commutative, this pointwise operation inherits both, and the all-zero function z(n) = 0 acts as an identity (adding it changes nothing). So far that is a commutative (Abelian) monoid. To be a group every function would need an inverse, i.e. a function -f with -f(n) = -(f(n)); but those values are negative whenever f(n) > 0, and negatives are not in A = {0,1,2,...}. So inverses are missing and (F, (-)) is an Abelian monoid, not a group.