# Concept derivation guides

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Concept guide



## [Metric space: auditing a finite distance table cell by cell and triple by triple](/en/derivations/derivation-mko-metric-space-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-metric-space)

Three complete integer distance tables are given. (A) On X={0,1,2,3}, table d (step distance on a 4-cycle): check nonnegativity, zero diagonal, nonzero off-diagonal entries and symmetry cell by cell, and the triangle inequality by a case argument covering all 4³=64 triples, concluding that (X,d) is a metric space. (B) Table e on Y={0,1,2} passes the first four checks but violates the triangle inequality at (0,1,2). (C) Table f on Z={0,1,2} is nonnegative, symmetric and zero on the diagonal, yet has an off-diagonal zero and violates separation.

Metric space

derivation-mko-metric-space-01





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