A metric on S is a function that assigns a non-negative real number to every pair of points in S, satisfying the following conditions:
d^2 and d^(1/2) are two different ways of representing distance in a metric. d^2 represents the squared distance between two points, while d^(1/2) represents the square root of the distance between two points. In other words, d^2 is the square of d^(1/2), so they both represent the same distance, but in different forms.
The choice of metric depends on the nature of your data and the specific application. Generally, d^(1/2) is more commonly used as it is easier to interpret and work with, but there may be cases where d^2 is more suitable. It is important to consider the properties of the metric and how they align with your specific needs.
Neither d^2 nor d^(1/2) can be considered a "better" metric as they both have their own advantages and limitations. It is important to carefully evaluate your data and the properties of each metric to determine which one would be more suitable for your specific application.
Yes, there are many other metrics that can be used on S. Some common examples include the Euclidean metric, Manhattan metric, and Chebyshev metric. The choice of metric depends on the specific needs and characteristics of your data and the application at hand.