1 (d) 1. Let S = C[0,1] be the set of real-valued continuous functionsdefined on the closed interval [0,1], where we define f+ g and fg,as usual, by (ƒ+ g)(x) = f(x) + g(x) and (fg) (x) = f(x)g(x). Let0 and I be the constant functions 0 and 1, respectively. Show that(a) (S, +,) is a commutative ring with unity.(b)S has nonzero zero divisors.(c) S has no idempotents #0,1.(d)Let a = [0,1]. Then the set T = {ƒE S\ſ(a) = 0) is a subringsuch that fg, gfE T for all ƒE T and g E S.

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Answered: (d) Let a € [0,1]. Then the set T = {ƒ¤…

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(d) Let a € [0,1]. Then the set T = {ƒ¤ S\ƒ(a) = 0) is a subring such that fg, gfE T for all ƒE T and g E S.

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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Question

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1 (d)

1. Let S = C[0,1] be the set of real-valued continuous functions
defined on the closed interval [0,1], where we define f+ g and fg,
as usual, by (ƒ+ g)(x) = f(x) + g(x) and (fg) (x) = f(x)g(x). Let
0 and I be the constant functions 0 and 1, respectively. Show that
(a) (S, +,) is a commutative ring with unity.
(b)
S has nonzero zero divisors.
(c) S has no idempotents #0,1.
(d)
Let a = [0,1]. Then the set T = {ƒE S\ſ(a) = 0) is a subring
such that fg, gfE T for all ƒE T and g E S.

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