Equivalence classes and Induced metric

[Lily@pie]

(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

 

[mathwonk]

triangle inequality? (i.e. re read the proof that ≈ is an equivalence relation.)

 

[Lily@pie]

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^*