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2. Let X be the reals and define p(x, y) = 2/x - y). Show that p is a pseudo- metric equivalent to the usual pseudometric for X.

2. Let X be the reals and define p(x, y) = 2/x - y). Show that p is a pseudo- metric equivalent to the usual pseudometric for X.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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the rational 2. Let X be the reals and define p(x, y) = 2x - y). Show that p is a pseudo- metric equivalent to the usual pseudometric for X. 3. Let X be the plane and for x = (1, 2) and y = (v₁.32), let p(x, y) = 1-₁+12-y2|- (a) Show that p is a pseudometric. (b) Describe the p-cell of radius r centered at the point (a, b). (e) Find the p-closure of S = {re X: r + x² < 1}.
Transcribed Image Text:the rational 2. Let X be the reals and define p(x, y) = 2x - y). Show that p is a pseudo- metric equivalent to the usual pseudometric for X. 3. Let X be the plane and for x = (1, 2) and y = (v₁.32), let p(x, y) = 1-₁+12-y2|- (a) Show that p is a pseudometric. (b) Describe the p-cell of radius r centered at the point (a, b). (e) Find the p-closure of S = {re X: r + x² < 1}.
Expert Solution
Step 1

To show that the function d(x,y) = 2|x-y| is a pseudometric on X, we need to show that it satisfies the following properties:

  1. Non-negativity: d(x,y) ≥ 0 for all x,y in X, and d(x,y) = 0 if and only if x = y.
  2. Symmetry: d(x,y) = d(y,x) for all x,y in X.
  3. Triangle inequality: d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z in X.

Note: we are taking d(x,y) instead of row(x,y) for convenience of writing the symbol

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