1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Metric space Triangular inequalities

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data

    let [tex]
    (X,\sigma) [/tex] be a metric space. [tex] xyz \in R[/tex]show that
    [tex]
    \mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)[/tex]

    2. Relevant equations



    3. The attempt at a solution

    [tex]\mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)=\mid \sigma(x,y) \mid = \mid\sigma
    (z,x) + \sigma(z,y) \mid[/tex]

    [tex] \leq \mid\sigma
    (x,z)\mid + \mid \sigma(y,z) \mid[/tex]


    Does that look alright?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 11, 2010 #2
    For real numbers a and b, |a+b| [tex]

    \leq
    [/tex] |a| + |b|.
    Thus, |a| + |b|.= |(a-c) + (c-b)| [tex]

    \leq
    [/tex] |a-c| + |c-b|
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook