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Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise addition f+g of by (f+g)(x) = f(x) + g(x), Vx R, and pointwise multiplication by (f g)(x) = f(x) g(x), Vx R. (a) Verify that the pair (M(R), +) form a group. (b) Verify that the pair (M(R), .) does NOT form a group. Suggest a modification to the underlying set M(R) (you may call this new set M(R)#), so that the pair (M(R)#, .) would form a group. (c) Define a set H(R) := {f = M(R) : f(x) = Z, for each x = R}. Prove that the pair (H(R), +) is a group.

Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise addition f+g of by (f+g)(x) = f(x) + g(x), Vx R, and pointwise multiplication by (f g)(x) = f(x) g(x), Vx R. (a) Verify that the pair (M(R), +) form a group. (b) Verify that the pair (M(R), .) does NOT form a group. Suggest a modification to the underlying set M(R) (you may call this new set M(R)#), so that the pair (M(R)#, .) would form a group. (c) Define a set H(R) := {f = M(R) : f(x) = Z, for each x = R}. Prove that the pair (H(R), +) is a group.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise addition f + g of by (f+g)(x) = f(x) + g(x), Vx € R, and pointwise multiplication by (f g)(x) = f(x) · g(x), Vx R. (a) Verify that the pair (M(R), +) form a group. (b) Verify that the pair (M(R), .) does NOT form a group. Suggest a modification to the underlying set M(R) (you may call this new set M(R)#), so that the pair (M(R)#, .) would form a group. E (c) Define a set H(R) := {f = M(R) : f(x) = Z, for each x € R}. Prove that the pair (H(R), +) is a group.
Transcribed Image Text:Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise addition f + g of by (f+g)(x) = f(x) + g(x), Vx € R, and pointwise multiplication by (f g)(x) = f(x) · g(x), Vx R. (a) Verify that the pair (M(R), +) form a group. (b) Verify that the pair (M(R), .) does NOT form a group. Suggest a modification to the underlying set M(R) (you may call this new set M(R)#), so that the pair (M(R)#, .) would form a group. E (c) Define a set H(R) := {f = M(R) : f(x) = Z, for each x € R}. Prove that the pair (H(R), +) is a group.
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