# Equivalence classes and Induced metric

by Lily@pie
Tags: classes, equivalence, induced, metric
 P: 109 $(X,\rho)$ is a pseudometric space Define: x~y if and only if $ρ(x,y)=0$ (It is shown that x~y is an equivalence relation) Ques: If $X^{*}$ is a set of equivalence classes under this relation, then $\rho(x,y)$ depends only on the equivalence classes of x and y and $\rho$ induces a metric on $X^{*}$. Attempt: I know that from the question, $X^{*}=$ {[a]; $a\in X$} where $[a]={x\in X;\rho(x,a)=0}$ But I don't know how to go about proving that $\rho(x,y)$ depends only on [x] and [y]. I know i need to prove that $\rho(x,y)$ only depends on the all the $c\in X$ such that $\rho(c,x)=\rho(c,y)=0$. But I just don't know where to start... Thanks
 HW Helper Sci Advisor P: 9,371 triangle inequality? (i.e. re read the proof that ≈ is an equivalence relation.)
 P: 109 From the triangle inequality, ρ(x,y) ≤ ρ(x,a) + ρ(a,y) I know that ρ(x,a) = 0 if $x \in [a]$ or $a \in [x]$ and ρ(a,y) = 0 if $y \in [a]$ or $a \in [y]$. And this shows that ρ(x,y) depends on [x] and [y] only? And how do I show it induces a metric on X*

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