Is {X, max(d,r)} or (X, min(d,r)) a Metric Space?

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Discussion Overview

The discussion revolves around whether the constructs {X, max(d, r)} and (X, min(d, r)) can be classified as metric spaces, given that (X, d) and (X, r) are already established metric spaces. Participants explore the properties required for these constructs to satisfy the conditions of a metric space, particularly focusing on the triangle inequality.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if {X, max(d, r)} is necessarily a metric space and seeks clarification on (X, min(d, r)).
  • Another participant suggests that verifying properties like d(x,y)=0 iff x=y and d(x, y)=d(y,x) is straightforward but expresses uncertainty about proving the triangle inequality.
  • A different participant proposes a potential form of the triangle inequality involving max functions but questions its validity under certain conditions.
  • Concerns are raised about the validity of the proposed inequality if r(x,z) exceeds d(x,z) while r(y,z) is less than d(y,z).
  • One participant acknowledges a typo in their previous post, indicating a collaborative effort to clarify points made.
  • Another participant expresses a belief that (X, min(d, r)) does not satisfy the conditions of a metric space, but this assertion is not universally agreed upon.

Areas of Agreement / Disagreement

Participants express differing views on the validity of (X, min(d, r)) as a metric space, with some suggesting it is not, while the status of {X, max(d, r)} remains under debate. The discussion does not reach a consensus.

Contextual Notes

Participants have not fully resolved the conditions under which the triangle inequality holds for the proposed metrics, and there are indications of missing assumptions regarding the relationships between d and r.

ag2ie
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If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?
 
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Well, what do you think?? What have you tried?
 


It's easy to verify d(x,y)=0 iff x=y and d(x, y)=d(y,x),
but I don't know how to prove triangle Inequality...
 


Well, is the following trye

[tex]d(x,z)\leq \max\{d(x,y),r(x,y)\}+\max\{d(y,z),r(y,z)\}[/tex]

?
 
Last edited:


yes..but if r(x,z) is greater than d(x,z), and r(y,z) is smaller than d(y,z), then this inequality is not necessary true...right?
 


Sorry, I made a typo, check the post again.
 


I see...if r(x,y) is greater than d( x,y), then d(x,z)≤max{d(x,y),r(x,y)}+max{d(y,z),r(y,z)} is also true...

Thanks ...and I think (X, min(d, r)) is not a metric space..right?
 


ag2ie said:
Thanks ...and I think (X, min(d, r)) is not a metric space..right?

right.
 


Thanks..you are really helpful
 

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