Given z1, z2 E C, define the distance between z1 and z2 by setting: d(z1, z2) := |21 – 22| . Prove the validity of the following properties: (i) d(z1, z2) > 0 for every z1, z2 E C, and d(z1, 22) = 0 if and only if z1 = 2. (ii) d(z1, z2) = d(z2, z1) for every z1, 22 E C. (iii) d(z1, z2) < d(z1, 23) + d(z3, 22) for every z1, 22, 23 E C.
Given z1, z2 E C, define the distance between z1 and z2 by setting: d(z1, z2) := |21 – 22| . Prove the validity of the following properties: (i) d(z1, z2) > 0 for every z1, z2 E C, and d(z1, 22) = 0 if and only if z1 = 2. (ii) d(z1, z2) = d(z2, z1) for every z1, 22 E C. (iii) d(z1, z2) < d(z1, 23) + d(z3, 22) for every z1, 22, 23 E C.
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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Transcribed Image Text:Given z1, z2 E C, define the distance between z1 and z2 by setting:
d(21, 22) := |21 – z2| .
Prove the validity of the following properties:
(i) d(21, 22) > 0 for every z1, 22 E C, and d(21, 22) = 0 if and only if z1
2.
(ii) d(z1, z2) = d(z2, z1) for every z1, 22 E C.
(iii) d(z1, z2) < d(z1, z3) + d(z3, 22) for every 21, 2, 23 E C.
Hint: use the properties of the modulus function.
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