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Math Help - Metric spaces

  1. #1
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    Question Metric spaces

    Here's the question :

    Suppose C is a circle and for each a,b belonging to C, define d(a,b) to be the distance along the line segment form a to b. Then d is a metric on C.

    Now i don't really understand the meaning of this statement :

    The fact that the route from a to b goes outside C is irrelevant ; a metric is a simply a function defined on ordered pairs and does not take into account any "route traveled" from one point to another.

    Are'nt we suppose to define the metric within the given set, which in this case would be the circle C ? Then why do we consider a route that is outside the set, at all? Also does this mean that we cannot define the distance from a to b as the minor arc or the major arc between them , because route does not matter ?

    I am a little confused. Please Help.
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  2. #2
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    Re: Metric spaces

    Hey mrmaaza123.

    I think what its asking you to do is to show that the distance function satisfies all the metric properties which are d(a,b) = d(b,a), d(a,b) >= 0 and d(a,b) = 0 iff a = b along with d(a,b+c) >= d(a,b) + d(a,c) which I think are the metric properties.
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  3. #3
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    Re: Metric spaces

    Yes that is alright. Showing that it is a metric space is not a problem. What i am a little confused about is the theoretical stuff that's written after the question. I can't see the relevance of that statement to the question. Thanks for your reply though.
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  4. #4
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    Re: Metric spaces

    Quote Originally Posted by mrmaaza123 View Post
    Here's the question :

    Suppose C is a circle and for each a,b belonging to C, define d(a,b) to be the distance along the line segment form a to b. Then d is a metric on C.
    So a and b are points on the circle C and you measure the "distance from a to b" along the straight line from a to b.
    Now i don't really understand the meaning of this statement :

    The fact that the route from a to b goes outside C is irrelevant ; a metric is a simply a function defined on ordered pairs and does not take into account any "route traveled" from one point to another.

    Are'nt we suppose to define the metric within the given set, which in this case would be the circle C ?
    Are you clear on what you mean by "'define the metric within the given set"? A metric, on a set, is, by definition, a function that assigns a number to every pair of points in the set. The two points have to be in the set- HOW you get that number is no relevant.

    Then why do we consider a route that is outside the set, at all? Also does this mean that we cannot define the distance from a to b as the minor arc or the major arc between them , because route does not matter ?
    No, it does not mean that at all. We can certainly define a "distance" (metric) that way (and, of course, verify that the requirements for a "metric" are satisfied). Those are just not the only ways that a "metric" can be defined.

    I am a little confused. Please Help.
    I suggest that you review the definition of "metric" and look at more examples.
    A metric on set A is any function that assigns to every member of AxA (i.e ordered pairs of members of A) with the properties that:
    1) d(x, y)\ge 0 and d(x,y)= 0 if and only if x= y.
    2) d(x, y)= d(y, x).
    3) for any 3 points, x, y, and z, d(x, y)\le d(x, z)+ d(z, y).

    If such a function satisfies those conditions, no matter how it may be defined is a "metric".
    A simple example is, taking the set A to contain only the points "x" and "y" is d(x,y)= 1, d(x, x)= 0, d(y, y)= 0. In fact, given any set, the function that assigns "1" to any distinct points, and assigns "0" to (x, x) (where x is any member of the set) is a metric.
    Thanks from mrmaaza123
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