# Is d(x,y) = |x^3 - y^3| a Valid Metric in X?

• mynameisfunk
In summary: Therefore, d(x,y) satisfies the triangle inequality and is a metric in X.In summary, the given function d(x,y) = |x^3 - y^3| satisfies the three conditions for being a metric in X, including the triangle inequality. This can be proven by using the fact that |a + b| <= |a| + |b| and applying it to the given function.
mynameisfunk
Prove or disprove d is a metrix in $$X$$:
$$d(x,y)=|x^3-y^3|$$

OK, 3 conditions to meet:

(i) $$d(x,y)>0$$
(ii) $$d(x,y)=0$$ iff $$x=y$$
(iii) $$d(x,y) \leq d(x,r) + d(r,y)$$ for any $$r \epsilon X$$

the first 2 are obvious and I have solved this by proving all of the cases:

$$r \leq x \leq y$$, $$x \leq r \leq y$$, etc.

My problem is that I know there is a better proof that is much shorter. My professor did it in class but I still had gotten the problem correct so didnt write it down. Any suggestions on a simpler way to do this?

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iii is applying the triangle inequality for real numbers, which is a metric. here is a proof of that: http://math.ucsd.edu/~wgarner/math4c/derivations/other/triangleinequal.htm

Last edited by a moderator:
I think joshyxc1979 has misunderstood the question. The link he provides gives a proof that the usual metric, d(x,y)= |x- y|, satisfies the triangle inequality. That says nothing about whether this new metric satisfies it.

mynameisfunk, try using the fact that [itex]|x^2- y^3|= |x- y||x^2+ xy+ y^2|[itex].

ok.
$$|x^3-y^3| \leq |x^3-r^3| + |r^3-y^3|$$ gives
$$|x-y||x^2+xy+xy^2| \leq |x-r||x^2+xr+r^2| + |r-y||r^2+ry+y^2|$$
since we know the usual metric |x-y| holds, we need to show that,
$$|x^2+xy+y^2| \leq |x^2+xr+r^2| + |r^2+ry+y^2|$$
Now I am stuck, I am not sure If I am allowed to turn this into
$$|xy| \leq |xr|+|ry|$$?
If I did, this wouldn't eve hold true. Also, since it is not true, I cannot add $$|xy| |xr| |ry|$$ to the terms to get squares...

For anyone still looking for the solution:

|a + b| <= |a| + |b|

d(x,y) = |x^3 + y^3| = |(x^3 - r^3) + (r^3 - y^3)| <= |x^3 - r^3| + |r^3 - y^3| =
d(x,r) + d(r,y)

## 1. What is a metric in mathematics?

A metric is a mathematical function that measures the distance between two points in a given space. It is a fundamental concept in mathematics and is used to define the geometry of a space.

## 2. What is the significance of proving d(x,y) is a metric in a space X?

Proving that d(x,y) is a metric in a space X provides a mathematical framework for understanding the properties of that space. It allows us to define and measure distances between points, and to analyze relationships and patterns within the space.

## 3. How is d(x,y) defined as a metric in a space X?

In order for d(x,y) to be considered a metric in a space X, it must satisfy three key properties: the distance between two points must be non-negative, the distance between a point and itself must be zero, and the distance between two points must satisfy the triangle inequality. These properties ensure that d(x,y) behaves consistently as a measurement of distance in the space X.

## 4. What are some examples of metrics in various spaces?

In Euclidean space, the most commonly used metric is the Euclidean distance, which measures the straight-line distance between two points. In graph theory, the shortest path between two points is often used as a metric. Other examples of metrics include the Hamming distance in computer science and the Manhattan distance in urban planning.

## 5. How is proving d(x,y) is a metric in X related to real-world applications?

The concept of a metric is widely applicable in various fields, such as physics, computer science, and economics. For example, in physics, metrics are used to measure the distance between objects in space. In computer science, metrics are used to measure the efficiency of algorithms. In economics, metrics are used to measure economic indicators such as inflation and unemployment rates. Proving d(x,y) is a metric in X allows for the application of mathematical principles to real-world problems and scenarios.

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