Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Solve number 2
1. Let M be a set with two members a and b. Define the function D: M X M as follows: D(a, a) = D(b, b) 0, D (a, b) D(b, a) = = r where r is a positive real number. Prove that (M, D) is a metric space. = 2. Let M be a set with three elements: a, b and c. Define D: M X M as follows: D(x,x) = 0 for all x E M; D(x, y) = D(y, x) is a positive real number for xy. Say D(a, b) = r, D(a, c) = s, D(b, c) = t with r≤st. Prove that (M, D) is a metric space if and only if t ≤r + s.
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Transcribed Image Text:1. Let M be a set with two members a and b. Define the function D: M X M as follows: D(a, a) = D(b, b) 0, D (a, b) D(b, a) = = r where r is a positive real number. Prove that (M, D) is a metric space. = 2. Let M be a set with three elements: a, b and c. Define D: M X M as follows: D(x,x) = 0 for all x E M; D(x, y) = D(y, x) is a positive real number for xy. Say D(a, b) = r, D(a, c) = s, D(b, c) = t with r≤st. Prove that (M, D) is a metric space if and only if t ≤r + s.
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