Metrics on sets. Let each d : S × S R below be a distance function on pairs of s
ID: 3665320 • Letter: M
Question
Metrics on sets. Let each d : S × S R below be a distance function on pairs of sets from space S. For each of the functions below, either prove it is a metric or provide a counterexample.
e) d5(A,B)= ( |AB|^p + |BA|^p) ^1/p ,p1
f) d6(A,B) = ( |AB|^p + |BA|^p)^1/p / |AB| , p 1
Explanation / Answer
There are many types of metrics like Acoustic metrics, Complete metrics, Similarity metrics, String metrics, Pseudo metrics, Quasimetrics, Metametrics, semimetrics, and premetrics The metric function or distance function gives a definition for the distance between a couple of elements belonging to a set. Such set comprising a metric is called as the metric space. Metric comprises the topology on the set The inverse is not true in the sense that not every topology need to be generated bya metric space (e) d5(A,B) = (|A-B|^p + |B-A|^p) ^1/p , p>=1 plugging in the following values for variables, A = 3, B = 4, p = 2 d5(3,4) = (|3-4|^2 + |4-3|^2) ^1/2 = (1 + 1 ) ^ ½ = 2 ^ ½ = 1.41421356237 hence d5(A,B) = 1.41421356237 d6(A,B) = ( |AB|^p + |BA|^p)^1/p / |AB| , p 1 Let A = 7, B = 5, p = 2 plugging in values, d6(7,5) = (|7-5|^p + |5-7|^2) ^ ½ / |7 U 5| = 2 ^ 2 + 2 ^ 2 ) ^ ½ / 7 U 5 = 4 + 4 ) ^ ½ / 7 U 5 = 8 ^ ½ / 7 U 5 = 2.82842712475 / 7 U 5 hence d6(A,B) = 2.82842712475 / 7 U 5