Question

If d_1 : X × X → ℝ^+ and d_2 : X × X → ℝ^+ are distance metrics, then so is:

a.d_1 - d_2
b.d_1 + d_2
c.None of the above
d.Both of the above

Answer 1

The function d_1 + d_2 where d_1 and d_2 are distance metrics is also a distance metric, whereas d_1 - d_2 is not, as it may result in negative values. Therefore, the correct option is b. d_1 + d_2.

Step-by-step explanation:

The student has asked whether the functions d_1 : X \u00d7 X \u2192 \u211d^+ and d_2 : X \u00d7 X \u2192 \u211d^+, when combined through subtraction or addition, also result in a distance metric. Let's analyze the options given:

• d_1 + d_2: The sum of two metrics is indeed a metric. If both d_1 and d_2 satisfy the conditions of a metric (non-negativity, identity of indiscernibles, symmetry, and triangle inequality), their sum will also satisfy these conditions.
• d_1 - d_2: The difference between two metrics is not guaranteed to be a metric. The result could be negative, which would violate the non-negativity condition required for a metric. Therefore, d_1 - d_2 is not a metric.

Consequently, the correct answer is b. d_1 + d_2.