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Question 1 Which of the following is false? (A) For all x, y, z € M, \d(x, y) — d(x, z)| ≤d(z,y) (B) If r, y, z € M and d(x, y) > d(r, z), then y> z (C) For all x, y, z € M, d(x, y) -d(z,y) ≤d(z, x) (D) For all z, y € M, d(x, y) #0 (r‡y)

Question 1 Which of the following is false? (A) For all x, y, z € M, \d(x, y) — d(x, z)| ≤d(z,y) (B) If r, y, z € M and d(x, y) > d(r, z), then y> z (C) For all x, y, z € M, d(x, y) -d(z,y) ≤d(z, x) (D) For all z, y € M, d(x, y) #0 (r‡y)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 4TFE: True or False Label each of the following statements as either true or false. If xy=xz for all x,y,...
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Question 1 Which of the following is false? (A) For all x, y, z € M, \d(x,y) — d(x, z)| ≤d(z,y) (B) If x, y, z € M and d(x, y) > d(x, z), then y > z (C) For all x, y, z € M, d(x,y) - d(z,y) ≤d(z, x) (D) For all x, y = M, d(x, y) #0 ⇒ (x + y) Question 2 Let M and Y be metric spaces. Which of the following is false? (A) If f: M→Y is continuous and bijective, then M is closed if Y is compact (B) Every closed subset of a compact metric space is compact (C) For all x, y = M, there exists some r>0 such that y = B(x, ) (D) Every subset of three elements of M is compact Question 3 Let A be a subset of M. Which of the following is false? (A) The interior of the interior of A contains a neighborhood of each of its points. (B) The image of a compact set under a Lipschitz map is a closed set (C) Every contraction map on M is continuous on the whole of M. (D) If A is closed, then it has at least one accumulation point.
Transcribed Image Text:Question 1 Which of the following is false? (A) For all x, y, z € M, \d(x,y) — d(x, z)| ≤d(z,y) (B) If x, y, z € M and d(x, y) > d(x, z), then y > z (C) For all x, y, z € M, d(x,y) - d(z,y) ≤d(z, x) (D) For all x, y = M, d(x, y) #0 ⇒ (x + y) Question 2 Let M and Y be metric spaces. Which of the following is false? (A) If f: M→Y is continuous and bijective, then M is closed if Y is compact (B) Every closed subset of a compact metric space is compact (C) For all x, y = M, there exists some r>0 such that y = B(x, ) (D) Every subset of three elements of M is compact Question 3 Let A be a subset of M. Which of the following is false? (A) The interior of the interior of A contains a neighborhood of each of its points. (B) The image of a compact set under a Lipschitz map is a closed set (C) Every contraction map on M is continuous on the whole of M. (D) If A is closed, then it has at least one accumulation point.
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