Homework 1

Read this paper, which summarizes what we covered in the first class. Then, consider the versions v as belonging to a metric space (M, d).

1. Prove that the quality of a version, as defined in equation (1) on page 4 of the paper, is always in the interval [-1, +1]. You will have to use the definition of a metric to show this (let's hope that no-one vandalizes the definition while the homework is due!).

2. As you can see in the definition of a metric, a metric has to obey, for all x, y, z in the metric space:

• d(x, x) = 0
• d(x, y) >= 0 (non-negativity)
• d(x, y) = 0 iff x = y (identity of indiscernibles)
• d(x, y) = d(y, x) (symmetry)
• d(x, z) <= d(x, y) + d(y, z) (triangle inequality)

If you drop symmetry, you get the definition of a quasimetric.

If you drop identity of indiscernibles you get a pseudometric.

If you drop the triangle inequality, you get a semimetric.

Of course, you can also drop more than one requirement at once, even though admittedly semiquasipseudometrics (also known quasipseudosemimetrics) are not that interesting, in spite of their name ("Well, you see, I am writing a paper on quasipseudosemimetric spaces. And you are working on, some kind of flying bug, I understand?").

For each of these colorfully named variations of metric spaces, either prove that q \in [-1, +1], or provide a counterexample.

Turn in your homework in class, on paper. How old fashioned, I know.