3. Let M be an arbitrary set and define D D(x,x) = 0 for all x EM; for xy, D(x, y) te [1, 2]. prove that (M, D) is a metric space. = M x M as follows: D(y,x)= t where
3. Let M be an arbitrary set and define D D(x,x) = 0 for all x EM; for xy, D(x, y) te [1, 2]. prove that (M, D) is a metric space. = M x M as follows: D(y,x)= t where
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve number 3
![3. Let M be an arbitrary set and define D M x M as follows:
D(x,x) = 0 for all x = M; for xy, D(x, y) = D(y,x): = t where
te [1, 2]. prove that (M, D) is a metric space.
4. Let M be a set with a real valued function D M x M satisfying
the following:
(1) D(a, a) = 0;
(2) D(a, b) #0 for a b;
(3) D(a, b) + D(b, c) ≥ D(c, a) for all a, b,and c.
Prove that (M, D) is a metric space. (Note: It is not assumed that
D(a, b) ≥ 0 and D(a, b) = D(b, a). You need to prove them.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeb81122-6fbf-4c63-aaf5-71689ff4fea7%2F38416202-8e86-4886-aaba-b94a56cb116d%2Fqfihcr3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let M be an arbitrary set and define D M x M as follows:
D(x,x) = 0 for all x = M; for xy, D(x, y) = D(y,x): = t where
te [1, 2]. prove that (M, D) is a metric space.
4. Let M be a set with a real valued function D M x M satisfying
the following:
(1) D(a, a) = 0;
(2) D(a, b) #0 for a b;
(3) D(a, b) + D(b, c) ≥ D(c, a) for all a, b,and c.
Prove that (M, D) is a metric space. (Note: It is not assumed that
D(a, b) ≥ 0 and D(a, b) = D(b, a). You need to prove them.)
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