500hw6 - SYMMETRY suppose then , but is a metric on so: ,...

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2 a) 2 b )
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1 a) 1 b) 3 a) Let (X,d) be a metric space and let D:X*X-->R be defined by: D(x,y) = (0 if x=y), (max(1,d(x,y)) otherwise)
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Prove that (X,D) is a metric space. I know that I have to show (X,D) is positive definite, symmetric and satisfies the triangle inequality. The first I am comfortable with. However, I am not sure of how to go about showing symmetry for the x/=y case. ..same for the triangle inequality. Anybody have any hints or suggestions?
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Unformatted text preview: SYMMETRY suppose then , but is a metric on so: , so: . If then Hence D is symmetric in its arguments (and this works whether or is used in the definition of ). TRIANGLE INEQUALITY Suppose then . Suppose: , then but is a metric on , so: , for any . So . Hence: . Now suppose: , then , but for all , so: . for any . When so the triangle inequality holds trivialy....
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500hw6 - SYMMETRY suppose then , but is a metric on so: ,...

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