Equivalence classes and Induced metric


by Lily@pie
Tags: classes, equivalence, induced, metric
Lily@pie
Lily@pie is offline
#1
Mar24-12, 12:53 PM
P: 109
[itex](X,\rho)[/itex] is a pseudometric space

Define:
x~y if and only if [itex]ρ(x,y)=0[/itex]
(It is shown that x~y is an equivalence relation)

Ques:
If [itex]X^{*}[/itex] is a set of equivalence classes under this relation, then [itex]\rho(x,y)[/itex] depends only on the equivalence classes of x and y and [itex]\rho[/itex] induces a metric on [itex]X^{*}[/itex].

Attempt:
I know that from the question,


[itex]X^{*}=[/itex] {[a]; [itex]a\in X[/itex]} where [itex][a]={x\in X;\rho(x,a)=0}[/itex]

But I don't know how to go about proving that [itex]\rho(x,y)[/itex] depends only on [x] and [y]. I know i need to prove that [itex]\rho(x,y)[/itex] only depends on the all the [itex]c\in X[/itex] such that [itex]\rho(c,x)=\rho(c,y)=0[/itex].

But I just don't know where to start...

Thanks
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mathwonk
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#2
Mar24-12, 01:46 PM
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triangle inequality? (i.e. re read the proof that ≈ is an equivalence relation.)
Lily@pie
Lily@pie is offline
#3
Mar25-12, 03:13 AM
P: 109
From the triangle inequality,

ρ(x,y) ≤ ρ(x,a) + ρ(a,y)

I know that ρ(x,a) = 0 if [itex] x \in [a] [/itex] or [itex] a \in [x] [/itex] and ρ(a,y) = 0 if [itex] y \in [a] [/itex] or [itex] a \in [y] [/itex]. 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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