Asvanced Calculus
12. Consider A = R'. Ifu, v E A, the Hamming distance is defined...
12. Consider A = R'. Ifu, v E A, the Hamming distance is defined by d(u, v) to be the number of coordinates in which they differ. For example if u = (0,1,2) and v = (0,5,6) then d(u, v) = 2 since the vectors differ in the 2nd and 3rd coordinate, but agree in the 1st. (a) Show that d(u, v) is a metric on A. (b) Let S be the subset of A consisting of the two points (0,0,0) and (1,1,1). Is San open set in (A,d)? Is it a closed set in (A, d)? (e) Let T = {(1/n, 0,0): n = 1,2,3,...}. Find the closure of T in (A,d). (d) BONUS: (This part is optional.) Extend the definition of Hamming distance to R" and prove by induction that d(u, v) is a metric