Problem 7 Let X be a non-empty set, and let d : X × X ⟶ R be a real-valued function which satisfies the following three conditions: d(x, y) ≥ 0, and x = y ⇒ d (x, y) = 0, d (x, y) = d(y, x), d(x, y) ≤ d(x, z) + d(z, y). A function d with these properties is called a pseudo-metric on X. A metric is obviously a pseudo-metric. (i) Give an example of a pseudo-metric which is not a metric. (ii) Let d be a pseudo-metric on X, define a relation ∼ in X by means of x ∼ y ⇔ d (x, y) = 0 and show that this is an equivalence relation whose corresponding set of equivalence classes can be made into a metric space in a natural way.