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from the actual definition of metric space ,we know that

metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality i am interested what is a symmetric distance?i know triangle equality,something sum of two length is more then third one,but what about symmetric distance?thanks in advance

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2 Answers 2

up vote 6 down vote accepted

Saying that the metric (or distance) is symmetric just means that the distance from $x$ to $y$ is always the same as the distance from $y$ to $x$. In symbols, for all $x,y\in X$ we have $$d(x,y)=d(y,x)\;.$$

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thanks in advance,thanks guys –  dato datuashvili Sep 18 '13 at 7:02
    
could there be non symmetric distance? –  dato datuashvili Sep 18 '13 at 7:03
    
@dato: People have looked at non-symmetric distance functions, because they have real-world applications. In terms of effort, for instance, the ‘distance’ up a hill is longer than the ‘distance’ down that hill if you’re a runner or cyclist. They’re a lot harder to work with, however, and they don’t have nearly so nice an associated body of theory. –  Brian M. Scott Sep 18 '13 at 7:05
    
but in terms of algebraic does there exist non symmetric distance? –  dato datuashvili Sep 18 '13 at 7:06
    
@dato: One can write down asymmetric functions that model the kind of psychological ‘distance’ that I mentioned in the other comment, but they aren’t usually thought of as distance functions; in my (limited) experience they’re more likely to be thought of as measuring energy expenditure, work done, or some psychological analogue of those. –  Brian M. Scott Sep 18 '13 at 7:09

That is, we can't have any $x,y$ in the set such that $d(x,y)>d(y,x)$. We must have $d(x,y)=d(y,x)$ for all $x,y$ in the set. (We want the distance from the one to the other to be the same as the distance from the other to the one, since that's how distance actually works "in real life.")

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