Question

(three parts): Thinking of R2 as the plane, consider the following three functions (R2)2- R: . de((a, b), (с, d))-V (a-c)2 + (b-d)2. The first is the usual distance formula (which we will take for granted to be a metric), the second is the "grid" distance (imagine you can only move horizontally or vertically). and the third is just comparing the distances on the line of the r-coordinates and of the y-coordinates and taking the maximum 1. Show that dsum and dmar are each metrics. Carefully showing that de is a metric is a bit messy. 2. Figure out what "circles" look like in each metric. (Precisely: draw the balls of radius 1 centered at the origin in the sense of each metric_in the notation from class, draw ((0,0, B^sum ((0. 0)), and Bi ((00) 3. Show that de, dsum, and dmar despite their clear differences each yield the same topology on R2.

Answer 1

D(x, y) = 0 iff squareroot (a - c)^2 + (b - d)^2 = 0 iff a - c = 0 & b - d = 0 iff a = C & b = d iff x = y d(x, y) = squareroot (a - c)^2 + (b - d)^2 = squareroot (c - a)^2 + (d - b)^2 = d (y, x) we take x = (x_1, x_2) y = (y_1, y_2) u = (u_1, u_2) } elementof R^2 u = (u_1, u_2) d(x, u) = squareroot (x_1 - u_1)^2 + (x_2 - u_2)^2 = squareroot (x_1 - y_1 + y_1 - u_4)^2 + (x_2 - y_2 + y_2 - u_2)^2 let us v_1 = x_1 - y_1 w_1 = y_1 - u_4 v_2 = x_2 - y_2 w_2 = y_2 - u_2 apply squareroot (v_1 + w_1)^2 + (v_2 + w_2)^2 lessthanorequalto squareroot v_1^2 + v_2^2 + squareroot w_1^2 + w_2^2 \r\n squareroot (x_1 - u_4)^2 + 9x_2 - u_2)^2 lessthanorequalto squareroot (x_1 - y_1)^2 + (x_2 - y_2)^2 + squareroot (y_1 u_4)^2 + (y_2 - u_2)^2