Hi, I was just thinking maybe we could help each other for the tests by listing the metrics and nonmetrics that we already know. That would be helpful as we already "recognize" them if ever we are asked to identify which is/is not a metric in a given list in a GRE (or any multiple choice) test.
Listed are what I can think of off hand:
1. $$d(x, y)=xy$$
2. $$d(x, y)=\min\{xy, 1\}$$ (is this true if 1 is replaced by any $$r\in\mathbb{R}$$?)
3. $$d(x, y)=0$$ if $$x=y$$ and $$1$$ otherwise. (can replace 1 by any $$r\in\mathbb{R}$$)
4. $$d(x, y)=d'(x, y)/(1+d'(x, y))$$ whenever $$d'(x, y)$$ is a known metric.
5. $$d(x, y)=xy/3$$ (can probably replace $$xy$$ by any known metric $$d'$$ or 3 by any real >0.
Here is an example of a nonmetric:
1. $$f(x, y)=(xy)^2$$
Please feel free to add to the list. Motivation behind the metric would also be useful, so is the area of mathematics where it crops up.
Please feel free to correct if there are mistakes.
Common Metrics
Re: Common Metrics
For 2 and 3, r > 0 works always.
For 5, you are correct, if d is a metric, kd is a metric any k > 0.
The common metrics in $$\mathbb{R}^n$$, where d any metric in $$\mathbb{R}$$
1. Euclidean $$\sqrt{\sum{(d(x_i, y_i))^2}}$$
2. Taxicab $$\sum{d(x_i, y_i)}$$
3. Sup $$\sup\{d(x_i, y_i)\}$$
For 5, you are correct, if d is a metric, kd is a metric any k > 0.
The common metrics in $$\mathbb{R}^n$$, where d any metric in $$\mathbb{R}$$
1. Euclidean $$\sqrt{\sum{(d(x_i, y_i))^2}}$$
2. Taxicab $$\sum{d(x_i, y_i)}$$
3. Sup $$\sup\{d(x_i, y_i)\}$$
Re: Common Metrics
This is also a metric in $$\mathbb{R}^n$$
$$S_p(\overline{x}, \overline{y}) = \displaystyle \sqrt[p]{\displaystyle \sum_{i = 1}^{n}(x_i  y_i)^p}$$ for $$p \in \mathbb{N}, p \geq 2$$
$$S_p(\overline{x}, \overline{y}) = \displaystyle \sqrt[p]{\displaystyle \sum_{i = 1}^{n}(x_i  y_i)^p}$$ for $$p \in \mathbb{N}, p \geq 2$$
Re: Common Metrics
Also, if d is a metric, then $$\sqrt{d}$$ is, too. However, $$d^2$$ might not be, as your first nonexample shows. I have a flashcard that asks me to differentiate between these two, as I often forget which is which.
Good idea with this topic, by the way. I wish there was a lot more analysis and topology on the exam so I could justify studying those topics more frequently. Wait, let me qualify that: I wish there was a lot more noncomplex analysis and topology on the exam.
Good idea with this topic, by the way. I wish there was a lot more analysis and topology on the exam so I could justify studying those topics more frequently. Wait, let me qualify that: I wish there was a lot more noncomplex analysis and topology on the exam.

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 Joined: Wed Jul 22, 2009 1:20 am
Re: Common Metrics
Most likely, there is at most one question on identifying metric. Being able to answer by a quick glance would be very helpful!
If there are two then it is great if each is answered by quick glance through. If the whole test is about identifying metric, then we finish the test in under 165 sec., assuming each question takes at most 2.5 sec. to answer. (Just a wild imagination... Haha...)
If there are two then it is great if each is answered by quick glance through. If the whole test is about identifying metric, then we finish the test in under 165 sec., assuming each question takes at most 2.5 sec. to answer. (Just a wild imagination... Haha...)