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 g : X →R. Show that your map g is well-defined, injective, and surjective. 2. Declare that a subset A of X is basic if A = Xn B,(r, y)) for some open ball B,((r,y)) = {(a, b) €R x R V(z – a)² + (y – b)? < e}. Let Aj and Az be basic subsets of X. i. Is Aj U Az a basic subset of X? ii. Is A, N Az a basic subset of X? 3. In a similar manner, declare that a subset I of R is basic if I=RN B.(x) for some open ball B,(r) = {a € R| |r – a| < e}. Given a basic subset A of X, is g(A) a basic subset of IR? Justify your answer.
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 g : X →R. Show that your map g is well-defined, injective, and surjective. 2. Declare that a subset A of X is basic if A = Xn B,(r, y)) for some open ball B,((r,y)) = {(a, b) €R x R V(z – a)² + (y – b)? < e}. Let Aj and Az be basic subsets of X. i. Is Aj U Az a basic subset of X? ii. Is A, N Az a basic subset of X? 3. In a similar manner, declare that a subset I of R is basic if I=RN B.(x) for some open ball B,(r) = {a € R| |r – a| < e}. Given a basic subset A of X, is g(A) a basic subset of IR? Justify your answer.
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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