Metric: Difference between revisions
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By symmetry, <math>d(x,y)=d(y,x)</math>. Thus | By symmetry, <math>d(x,y)=d(y,x)</math>. Thus | ||
<math display="block">\begin{align} | <math display="block">\begin{align} | ||
0 | &0\le d(x,y)+d(x,y)\\ | ||
0 | &0\le 2d(x,y)\\ | ||
0 | &0\le d(x,y) | ||
\end{align}</math>}} | \end{align}</math>}} | ||
Revision as of 20:02, 12 June 2026
In mathematics, a metric or distance function is a function that defines a rigorous concept of distance between each pair of elements in a given set. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as open and closed sets, continuity, convergence, and limits in highly abstract spaces. A set equipped with a metric is called a metric space.
Definitions
Let 
be a given non-empty set. A metirx on is a function
,
that satisfies the following four properties for all 
:
- Non-negativity:
- Identity of indiscernibles:
- Symmetry:
- Triangle inequality:
.
Properties
Property.
Non-negativity is implied by the other three properties.
Proof
By the identity of indisscernbles, 
. Thus
By symmetry, . Thus
Let 
. By triangle inequality,
. Thus
. Thus
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