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.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Solve number 4
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.)
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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