Answered: Let A be a set and d : A × A → {0, 1,…

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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I have the following question:

Let A be a set and d : A × A → {0, 1, 2} a symmetric set
function that satisfies d(x, y) = 0 ⇔ x = y. Show that (A, d) is a metric
space.

please if able write the proof in a detailed manner with some explanation. I seem to struggle with proving the obvious or proving something the wrong way like for example with the triangle inequality somehow it takes way to long to prove. Thank you in advance.

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Follow-up Question

thank you for the detailed answer, I have a few more questions, if the condition that d is symmetric wasn't given how would I approach proving that d is symmetric?

and is there a shorter way to prove the triangle inequality without going through all twelve cases?

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