Difference between revisions of "Metric (mathematics)"
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− | In mathematics a set is said to have a '''metric''' if there is a notion of distance on the set that fits our intuition about how distance should behave. More formally, an operation <math> d(x,y) </math> on a set '''A''' to the real numbers is said to be a metric if for all <math> x,y</math> and <math> z </math>, we have the following | + | In mathematics a set is said to have a '''metric''' if there is a notion of distance on the set that fits our intuition about how distance should behave. More formally, an operation <math> d(x,y) </math> on a set '''A''' to the real numbers is said to be a metric if for all <math> x,y</math> and <math> z </math>, we have the following four properties. |
− | # | + | # ''d''(''x'', ''y'') ≥ 0 (''non-negativity'') |
− | # | + | # ''d''(''x'', ''y'') = 0 if and only if ''x'' = ''y'' (''identity'') |
− | # | + | # ''d''(''x'', ''y'') = ''d''(''y'', ''x'') (''symmetry'') |
+ | # ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') (''[[triangle inequality]]''). | ||
− | Intuitively property 1 says that distance cannot be negative and two | + | Intuitively, property 1 says that distance cannot be negative and two elements of the set are zero distance away from each other if and only if they |
− | are | + | are the same element. Property 2 says that the distance from x to y is the same as the distance from y to x. Property three says that the distance going from x to y to z is at least the distance to go from x to z. This third property is known as the "triangle inequality." |
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
+ | [[Category:Topology]] |
Latest revision as of 09:59, 13 July 2016
In mathematics a set is said to have a metric if there is a notion of distance on the set that fits our intuition about how distance should behave. More formally, an operation on a set A to the real numbers is said to be a metric if for all and , we have the following four properties.
- d(x, y) ≥ 0 (non-negativity)
- d(x, y) = 0 if and only if x = y (identity)
- d(x, y) = d(y, x) (symmetry)
- d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
Intuitively, property 1 says that distance cannot be negative and two elements of the set are zero distance away from each other if and only if they are the same element. Property 2 says that the distance from x to y is the same as the distance from y to x. Property three says that the distance going from x to y to z is at least the distance to go from x to z. This third property is known as the "triangle inequality."