Answer 1-2 Let X be the graph of f(x) = r2/3 given belowthat is, X is the subset of R x R satisfying the given equation.1. Define a bijective map g : X → R. Show that your map g is well-defined, injective, and surjective.2. Declare that a subset A of X is basic ifA = Xn B,((r, y))for some open ball B,((r, y)) = {(a, b) € R x R| V(I – a)² + (y – b)? < e}.Let Aj and A2 be basic subsets of X.i. Is Aj U Az a basic subset of X?ii. Is A N A, a basic subset of X?3. In a similar manner, declare that a subset I of R is basic ifI = RN B(r)for some open ball B.(r) = {a € R| |r – a| < e}. Given a basic subset A of X, is g(A) a basicsubset of R? Justify your answer.

Answered: Let X be the graph of f(x) = r/3 given…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 20E: Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ]...

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Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
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Let X be the graph of f(x) = r/3 given below that is, X is the subset of R x R satisfying the given equation. 1. Define a bijective map g : XR. Show that your map g is well-defined, injective, and surjective.