Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm looking for the proof of that: d(x,y) = d (y,x). I know that I have to use the "non-negativity" and "triangle inequality" but I don't know how to combine them to get the result.

share|cite|improve this question

closed as off-topic by avid19, RecklessReckoner, Shailesh, Jendrik Stelzner, Michael Albanese Jan 22 at 2:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – avid19, Shailesh, Jendrik Stelzner
If this question can be reworded to fit the rules in the help center, please edit the question.

1  
In a metric space, this is part of the definition, so not something that can be proved. – Tobias Kildetoft Oct 8 '13 at 18:09
    
However, you may be asked to prove that some distance function $d$ is a metric, in which case you would need to verify the symmetry property. By the way, symmetry does not follow from nonnegativity plus triangle inequality; why do you think it does? – user43208 Oct 8 '13 at 18:12
    
I want to proove that the symmetry follows from nonnegativity and triangle inequality or not. – Alex Oct 8 '13 at 18:14
up vote 1 down vote accepted

Take $d(x,y)=x^2+2y^2$. We obviously have nonnegativity and $d(x,y)+d(y,z)=x^2+2y^2+y^2+2z^2\geq x^2+2z^2=d(x,z)$ so the triangle inequality holds. But, for instance, $d(1,0)=1$ and $d(0,1)=2$, so symmetry does not hold. So symmetry does not follow from nonnegativity and the triangle inequality.

As mentioned in the comments, symmetry is a part of the definition of a distance function. A distance function is a real-valued function which satisfies nonnegativity, symmetry, and the triangle inequality, as well as $d(x,y)=0\iff x=y$. You can come up with functions which satisfy any three of these and do not satisfy the fourth—that's why they're all required in the definition.

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.