Exercise. Consider the set R2. For every x =(x1, x2) E R? and y =(y1, y2) E R², writePi(x, y) =| x – y1 |+ | #2 – Y2 \,andP2(x, y) = max{| x1 – yn |,| x2 - y2 |}.Prove that (R², p1) and (IR?, p2) are metric spaces?

(a) Here, ρ1x, y=x1-y1+x1-y2 (1) show that ρ1x, y≥0 : We know that, x≥0, ∀x Therefore, x1-y1≥0,…

Exercise. Consider the set R. For every x = (x1, x2) E R? and y = (y1, y2) E R², write Pi(x, y) =| x – y1| + | #2 – Y2 l, - and p2(x, y) = max{| ¤1 – yı |, | 2 – y2 |}. Prove that (R2, p1) and (R2, p2) are metric spaces?

Exercise. Consider the set R. For every x = (x1, x2) E R? and y = (y1, y2) E R², write Pi(x, y) =| x – y1| + | #2 – Y2 l, - and p2(x, y) = max{| ¤1 – yı |, | 2 – y2 |}. Prove that (R2, p1) and (R2, p2) are metric spaces?

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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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Exercise. Consider the set R2. For every x =
(x1, x2) E R? and y =
(y1, y2) E R², write
Pi(x, y) =| x – y1 |+ | #2 – Y2 \,
and
P2(x, y) = max{| x1 – yn |,| x2 - y2 |}.
Prove that (R², p1) and (IR?, p2) are metric spaces?

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