The Round-Up Metric For all points on the .x-axis, represented conveniently by the coordin...

The Round-Up Metric For all points on the .x-axis, represented conveniently by the coordinates themselves, a concept for distance is defined as follows: d(x,y) = {the number | x - y| rounded up}, that is, either |x - y| itself if u is an integer, or the next higher integer if it is not. [Examples: d(2,5) = 3, d(2,5.5) = 4, and d(-2, 5.5) = 8.]

(a) Does this concept of distance satisfy Axioms D-1, D-2, and D-3? D-4?

(b) Identify the segment . (Hint: The answer is {2, 3, 4, 5} and no other points. Explain.)

(C) Identify the segment where a = 5.5.

(d) Identify the segment where a = 2.5 and b = 5.5. Generalize to arbitrary numbers a and b.

(e) Prove the triangle inequality for this metric.