a) Give an example of an infinitely large set A and a function d: A × A → R≥0 so that d satisfies the triangle inequality, but is degenerate (i.e. there exist several x, y ∈ A with d(x, y) = 0). (b) Give an example of an infinitely large set A and a function d: A × A → R≥0 so that d satisfies the triangle inequality, but is not symmetrical. (c) Give an example of an infinitely large set A and a function d: A×A → R≥0 so that d does satisfy the triangle inequality, but at the same time degenerate and not symmetrical.
a) Give an example of an infinitely large set A and a function d: A × A → R≥0 so that d satisfies the triangle inequality, but is degenerate (i.e. there exist several x, y ∈ A with d(x, y) = 0). (b) Give an example of an infinitely large set A and a function d: A × A → R≥0 so that d satisfies the triangle inequality, but is not symmetrical. (c) Give an example of an infinitely large set A and a function d: A×A → R≥0 so that d does satisfy the triangle inequality, but at the same time degenerate and not symmetrical.
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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Question
a) Give an example of an infinitely large set A and a function
d: A × A → R≥0 so that d satisfies the triangle inequality, but
is degenerate (i.e. there exist several x, y ∈ A with d(x, y) = 0).
(b) Give an example of an infinitely large set A and a function
d: A × A → R≥0 so that d satisfies the triangle inequality, but
is not symmetrical.
(c) Give an example of an infinitely large set A and a
function d: A×A → R≥0 so that d does satisfy the triangle inequality,
but at the same time degenerate and not symmetrical.
I always struggle with questions like this where I have to find a set, please if able provide some insight on how to approach this best. Thank you very much.
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