7. (a) Let X = R2, and for x, y E R1 define d(x,y) by
d(x, y) = d( (xi, x2)) (yi, y2)) = 'xi — yi I { Ix' I + lx2 — y2I + IA
if x2 = y2 , if otherwise.
Show that (X, d) is a metric space. (b) Let X= R, and for x, y E R define d(x,y) = Ix' + Ix — yl + ly1 when x 0 y, and d(x, y) = 0 when x = y. Show that d is a metric on R. Hint: (a) Let x = (xi, 3c2),y = (yl, y2) and z = (z1, z2) be in X. We note firstly that ixi — Yii C d(x, y). If x2 = y2, then d(x, y) = 'xi — yi I C Ix' — zi I+ Izi — Yi I d(x, z) + d(z, y). If x2 y2, then z2 cannot be equal to both x2 and y2; so assume z2 x2. Then
d(x,y) = 'xi' + lx2 — y2I + 1)/11 :5. Ix' I + lx2 — z2I + Iz2 — y21 + 1)/11
{ (14 + lx2 — z2I + Izi I) + Izi — yi I
(b) Straightforward.
if y2= if y2
Z2 )
Z2