description/proof of that for metric space, distance between subsets does not necessarily satisfy triangle inequality
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of distance between subsets on metric space.
Target Context
- The reader will have a description and a proof of the proposition that for a metric space, the distance between some subsets does not necessarily satisfy the triangle inequality.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the metric spaces }\}\)
\(S_1\): \(\subseteq M\)
\(S_2\): \(\subseteq M\)
\(S_3\): \(\subseteq M\)
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Statements:
Not necessarily "\(dist (S_1, S_2) \le dist (S_1, S_3) + dist (S_3, S_2)\)"
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2: Proof
Whole Strategy: Step 1: see a counterexample.
Step 1:
Let \(M = \mathbb{R}\) as the Euclidean metric space.
Let \(S_1 = B_{- 1.5, 1}\), \(S_2 = B_{1.5, 1}\), and \(S_3 = B_{0, 1}\).
\(dist (S_1, S_2) = 1\), \(dist (S_1, S_3) = 0\), and \(dist (S_3, S_2) = 0\).
So, "\(dist (S_1, S_2) = 1 \le 0 = dist (S_1, S_3) + dist (S_3, S_2)\)" does not hold.
