# definition of metric space

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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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 $$d(x,y)=d(y,x)\;.$$
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.")