Question

3. Let A, B be non-empty compact subsets of (X, d). The Hausdorff distance is $d_H(A, B) := \max\{\sup_{a \in A} \inf_{b \in B} d(a, b), \sup_{b \in B} \inf_{a \in A} d(a, b)\}$. a) Show that $d_H$ is a metric. b) Let A = [0, 1], B = [3, 5]. Find $d_H(A, B)$ in R. c) Find $d_H(A, B)$ for the same A, B in R with the post office metric $d(x, y) = |x| + |y|$ for $x \neq y$, $d(x, x) = 0.$

          3. Let A, B be non-empty compact subsets of (X, d). The Hausdorff distance is
$d_H(A, B) := \max\{\sup_{a \in A} \inf_{b \in B} d(a, b), \sup_{b \in B} \inf_{a \in A} d(a, b)\}$.
a) Show that $d_H$ is a metric.
b) Let A = [0, 1], B = [3, 5]. Find $d_H(A, B)$ in R.
c) Find $d_H(A, B)$ for the same A, B in R with the post office metric $d(x, y) = |x| + |y|$ for
$x \neq y$, $d(x, x) = 0.$

3. Let A, B be non-empty compact subsets of (X, d). The Hausdorff distance is
dH(A, B) := max{supa ∈ Ainfb ∈ B d(a, b), supb ∈ Binfa ∈ A d(a, b)}.
a) Show that dH is a metric.
b) Let A = [0, 1], B = [3, 5]. Find dH(A, B) in R.
c) Find dH(A, B) for the same A, B in R with the post office metric d(x, y) = |x| + |y| for
x ≠ y, d(x, x) = 0.

Added by Margaret R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

10/18/2022

Step 1

Step 1: To show that dH is a metric, we need to show that it satisfies the properties of a metric: Show more…

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3. In real analysis, or functional analysis, please briefly show your work explicitly and clearly. 3. Let A, B be non-empty compact subsets of (X, d). The Hausdorff distance is dH(A, B) := max{sup inf d(a, b), sup inf d(a, b)} a∈A b∈B b∈B a∈A a) Show that dH is a metric b) Let A = [0, 1], B = [3, 5]. Find dH(A, B) in R c) Find dH(A, B) for the same A, B in R with the post office metric d(x, y) = |x| + |y| for x, y, d(x, x) = 0.