Let F denote M(R) [R being the set of real numbers], the set of all mappings f:

Question

Let F denote M(R) [R being the set of real numbers], the set of all mappings f: R ? R , made into a group
Let

H = {f ? F : f (x) ? Z for each x ? R}.

Prove that H is a subgroup of F. State clearly the properties of that you use.

Explanation / Answer

To prove that H is a subgrp of F we must prove that the set is closed under the addition and inversion To prove closure we have if f is in H then f:R->Z then the inverse is -f : R-> Z wheer -f(x) =-f(x) then -f is the inverse of f as f+(-f)=0 To prove closure we have g,f in F then f+g = f(x)+g(x) as both f(x), g(x) are integers so is f+g(x) hence f+g is in H hence H is closed under +. Hence H is subgrp of F Hence proved

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