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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 30E

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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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