Let X = {[a, b] : a, b e R and a 0: [u, v] 5 [r , 8+ ]} (you may assume that this minimum exists) and use this function to define a distance function on X, d: X X X R, by setting d([r, s], [u, v]) = max{A([r, s], [u, v]), A([u, v], [r, s])}. page 2 of 6 (a) Find A([0, 1], [0,3]), A([0, 3], [0, 1]), d([0, 1], [0, 3]), d([0,0], [2,2]). (Note that the two intervals in the last example are sets containing single points.) (b) (i) Prove that d satisfies (M1). (ii) Prove that d satisfies (M2). (iii) Suppose that (u, v), [r, s) and (x, y) are in X. (1) Show that if di = A([u, v], [r, s]) and 82 = A([r, s], [x, y]), then Al[u, v], [x, y]) 0: [u, v] 5 [r , 8+ ]} (you may assume that this minimum exists) and use this function to define a distance function on X, d: X X X R, by setting d([r, s], [u, v]) = max{A([r, s], [u, v]), A([u, v], [r, s])}. page 2 of 6 (a) Find A([0, 1], [0,3]), A([0, 3], [0, 1]), d([0, 1], [0, 3]), d([0,0], [2,2]). (Note that the two intervals in the last example are sets containing single points.) (b) (i) Prove that d satisfies (M1). (ii) Prove that d satisfies (M2). (iii) Suppose that (u, v), [r, s) and (x, y) are in X. (1) Show that if di = A([u, v], [r, s]) and 82 = A([r, s], [x, y]), then Al[u, v], [x, y])