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for (i) I know that the square root part is true but I don't know how to put it into words to prove it.

For (ii) I just don't know how top apply the requirements for a metric space to the square root of another metric space. Just kind of confusing me

Daniel
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  • i think i figured out the second part, i just defined the rho metric space to be |X-Y| and then can just go through the definition of a metric space can't I? – Daniel Dec 01 '15 at 03:23
  • No, the rho metric needs to be any possible metric. The |x -y| metric is only one and it's only possible if M is a subset of R which M probably is not. – fleablood Dec 01 '15 at 03:27

2 Answers2

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The prove it's a metric you need to show that

0) $\sqrt{\rho(x,y)}$ is a non-negative real value function.

1) $\sqrt{\rho(x,y)} = 0 \iff x = y$

2) $\sqrt{\rho(x,y)} = \sqrt{\rho(y,x)}$ for all $x, y \in M$.

3) $\sqrt{\rho(x,z)} \le \sqrt{\rho(x,y)} + \sqrt{\rho(y, z)}$ for all $x, y, z \in M$.

0) As $\rho(x,y)$ is a non-negative real function (as it is a metric) and the square root of any non-negative real number is also a non-negative real number. So $\sqrt{\rho(x,z)} $ is a non-negative real function.

1) $\sqrt{v} = 0 \iff v = 0$. $\rho(x,y) = 0 \iff x = y$. Therefore $\sqrt{\rho(x,y)} = 0 \iff \rho(x,y) = 0 \iff x = y$.

2 and 3 I leave to you.

fleablood
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Suppose to the contrary that $\sqrt{a}\gt \sqrt{b}+\sqrt{c}$. Then $$a\gt (\sqrt{b}+\sqrt{c})^2=b+c+2\sqrt{bc}\ge b+c,$$ so $a\gt b+c$, contradicting the fact that $a\le b+c$.

André Nicolas
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  • OK I was think about squaring both sides and that confirmed my first thought! Makes a lot more sense! Thank you! – Daniel Dec 01 '15 at 03:37
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    You are welcome. Sometimes simple algebra moves are enough! – André Nicolas Dec 01 '15 at 03:38
  • With the classes I've been taking lately I feel like using algebra is way to easy and that the problem is more complicated haha – Daniel Dec 01 '15 at 03:39
  • The hard part and the point of these exercises is pounding home the definitions. Once you hearing "phi is a metric" is as clear and as meaningful as "xhi is larger than psi" this is easier then algebra. But until then it's very frustrating and one is prone to over thinking. – fleablood Dec 01 '15 at 03:45