#2need a b c d e f gtrue or false with explanation Question 2Which of the following are true or false? If false, give examples. If true,motivate why they are true.(a) Let D C R be open and f : D → R° be continuous. Let G C Rº beopen and connected. Then f-1 (G) is open and connected.(b) Let D = {z € R" : |r|| < 1}. Let f : D → R° be continuous. Then f (D)contains all its cluster points.(c) Let D = {r € R" : ||-|| < 1} U {z € R" : 1< ||r|| < 2}. Let f : D→ R°be continuous. Then f (D) is disconnected.(d) Let A, C R' for j 2 1. If g is a cluster point of UA, then it is a clusterj=1point of A, for at least one j.(e) Let {F. : a € I} be a collection of closed subsets of R". Then R"\N F.contains only its interior points.(f) Let f : [a, b] → R be a strictly increasing function (so that r < y =f (1) < f (y)). Let e > 0. Then there is a polynomial P such that for allx € [a, b] ,|f (1) – P (r)| < E.(g) Let A be a non-empty connected subset of the Cantor set. Then Aconsists of just a single point.

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Question 2 Which of the following are true or false? If false, give examples. If true, motivate why they are true. (a) Let DC R be open and f: D- R° be continuous. Let G C R° be open and connected. Then f-' (G) is open and connected. (b) Let D = {z€ R" : ||-| < 1}. Let f : D → R° be continuous. Then f (D) contains all its cluster points. (c) Let D = {1 € R' : ||r|| < 1} U {z € R° : 1 < ||r|| < 2}. Let f : D R be continuous. Then f (D) is disconnected.

Question 2 Which of the following are true or false? If false, give examples. If true, motivate why they are true. (a) Let DC R be open and f: D- R° be continuous. Let G C R° be open and connected. Then f-' (G) is open and connected. (b) Let D = {z€ R" : ||-| < 1}. Let f : D → R° be continuous. Then f (D) contains all its cluster points. (c) Let D = {1 € R' : ||r|| < 1} U {z € R° : 1 < ||r|| < 2}. Let f : D R be continuous. Then f (D) is disconnected.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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