Equivalence classes and Induced metric

In summary, (X,\rho) is a pseudometric space where x~y if and only if ρ(x,y)=0. It has been shown that x~y is an equivalence relation. If X^{*} is a set of equivalence classes under this relation, then ρ(x,y) depends only on the equivalence classes of x and y. This is proven by the triangle inequality and the fact that ρ(x,y) = 0 for all a such that x \in [a] and y \in [a]. Furthermore, ρ(x,y) satisfies the properties of a metric on X^{*}, making it a true metric space.
  • #1
Lily@pie
109
0
[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
 
Last edited:
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  • #2
triangle inequality? (i.e. re read the proof that ≈ is an equivalence relation.)
 
  • #3
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*
 

FAQ: Equivalence classes and Induced metric

What are equivalence classes in mathematics?

Equivalence classes are sets of elements that are considered equivalent according to a specific relation. They are used to group elements that share similar properties or characteristics.

How are equivalence classes related to induced metric?

Equivalence classes can be used to define an induced metric on a set. This is done by considering the distance between two elements in an equivalence class to be the distance between their respective equivalence classes in the original set.

What is the purpose of induced metric?

The induced metric allows us to define a distance function on a set that may not have a natural metric. It is useful in situations where we want to measure the distance between elements in a set that do not have a direct way of measuring distance.

How is the induced metric calculated?

The induced metric is calculated by taking the distance between two elements in the original set and dividing it by the distance between their respective equivalence classes. This results in a new metric that reflects the properties of the original set and the equivalence relation.

What are some real-world applications of equivalence classes and induced metric?

Equivalence classes and induced metric are used in various fields such as graph theory, data analysis, and machine learning. They are also applied in areas such as social sciences and linguistics to group similar data or concepts together.

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