answer item 1 only. 1. Let X be the graph of f(r) = r 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. 4)) = {(a,b) € R x R| V - a)² + (y – b)? < c}.Let Aj and A, be basic subsets of X.i. Is Aj U Az a basic subset of X?ii. Is A, nA, a basic subset of X?3 In a similar manner, declare that a subset / 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 Y, is g(A) a basicsubset of R? Justify your answer.

Answered: 1. Let X be the graph of f(r) = r 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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1. Let X be the graph of f(r) = r given below that is, X is the subset of R x R satisfying the given equation. 1 Define a bijective map y : X → R. Show that your map g is well-defined, injective, and rurioctivn