Getting acquainted with metrics

For the set \(X\) and function \(\rho\colon X^2 \to \mathbb R\) determine whether the function \(\rho\) is a metric.

Variant 1
\(X = \{1, 2, 3, 4, 5\}\) a \(\rho\) is given by the table:

1 2 3 4 5

1 0 2 6 4 4

2 2 0 3 2 4

3 6 3 0 2 4

4 4 2 2 0 4

5 4 4 4 4 0
Resolution
No, since \(\rho(1, 2) + \rho(2, 3) < \rho(1, 3)\).
Variant 2
\(X = \{1, 2, 3, 4, 5\}\) a \(\rho\) is given by the table:

1 2 3 4 5

1 0 5 6 5 3

2 5 0 5 8 4

3 6 5 0 5 3

4 5 8 5 0 4

5 3 4 3 4 0
Resolution
Yes, the function is a metric. It is easy to manually verify that all of the metric axioms are met.

However, there is a more straightforward solution if we refer to the the Euclidean metric. We assign points to \( X \) points in \( R ^ 2 \) as follows: \(1 \to (0, 3)\), \(2 \to (4, 0)\), \(3 \to (0, -3)\), \(4 \to (-4, 0)\), \(5 \to (0, 0)\). Then the values in the table are exactly the distances between the points corresponding to the assigned points.