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Math Help - metrics

  1. #1
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    metrics

    Supoose that f -> X where D is any set and x is any metric space with metric d. We can define a function d*xD->R given by:
    d*(x,y) = d(f(x),f(y)) for any x,y in D. If f is injective prove that d* is a metric on D!

    Any help would be great!
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  2. #2
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    sorry its f: D -> X
    and later on d*: DxD -> R.
    the weird smiley faces come sometimes..
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  3. #3
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    Quote Originally Posted by really.smarty View Post
    Supoose that f -> X where D is any set and x is any metric space with metric d. We can define a function d*xD->R given by:
    d*(x,y) = d(f(x),f(y)) for any x,y in D. If f is injective prove that d* is a metric on D!
    There is really one of the three properties that is not trival.
    If d^*(x,y)=0 that mean d(f(x),f(y))=0.
    Does that mean f(x)=f(y)? And what does that mean?
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  4. #4
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    if f(x) = f(y) that means the first three parts are satisfied for metrics.
    1. not negative.
    2. d(x,y) = 0 iff x=y
    3. d(x,y) = d(y,x)

    i am more concerned with how to show the triangle inequality for this question?
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  5. #5
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    Quote Originally Posted by really.smarty View Post
    i am more concerned with how to show the triangle inequality for this question?
    d^* (x,y) = d(f(x),f(y)) \leqslant d(f(x),f(z)) + d(f(z),f(y))
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  6. #6
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    and i would go about proving that statement how?

    than you again.
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