a. There are several alternative ways to define a distance between two points p1(x1, y1)...

a. There are several alternative ways to define a distance between two points p₁(x₁, y₁) and p₂(x₂, y₂) in the Cartesian plane. In particular, the Manhattan distance is defined as

Prove that d_M satisfies the following axioms, which every distance function must satisfy:

b. Sketch all the points in the Cartesian plane whose Manhattan distance to the origin (0, 0) is equal to 1. Do the same for the Euclidean distance.

c. True or false: A solution to the closest-pair problem does not depend on which of the two metrics—d_E (Euclidean) or d_M (Manhattan)—is used?