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 three properties.  
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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 four properties.
  
# <math> d(x, y) \geq 0 </math> and this equality is strict if and only iff <math> x \neq y </math>
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# ''d''(''x'', ''y'') 0 &nbsp;&nbsp;&nbsp; (''non-negativity'')
# <math> d(x,y) = d(y,x) </math>
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# ''d''(''x'', ''y'') = 0 &nbsp; if and only if &nbsp; ''x'' = ''y'' &nbsp;&nbsp;&nbsp; (''identity'')
# <math> d(x, z) \leq d(x, y) + d(y, z) </math>
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# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')  &nbsp;&nbsp;&nbsp; (''symmetry'')
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# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') &nbsp;&nbsp;&nbsp; (''[[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  
 
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  
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[[Category:Mathematics]]
 
[[Category:Mathematics]]
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[[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.

  1. d(x, y) ≥ 0     (non-negativity)
  2. d(x, y) = 0   if and only if   x = y     (identity)
  3. d(x, y) = d(y, x)     (symmetry)
  4. 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."