What are the properties of distance?

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In summary, we discussed the properties of the distance function $d(u, w)$ and $d(V, W)$, defined as the Euclidean norm and the minimum distance between points in two sets $u, w \in \mathbb{R}^n$ and $V, W \subseteq \mathbb{R}^n$, respectively. We also mentioned that the definition of $d(V, W)$ can be problematic and may not always have a minimum value. We then discussed the definition of orthogonality, norm, and metric, and how these concepts relate to the properties of $d(u, w)$ and $d(V, W)$. Finally, we noted that there are other types of metrics besides the
  • #1
mathmari
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Hey!

Let $v, w\in \mathbb{R} ^n$ and let $V, W\subseteq \mathbb{R} ^n$.

I want to show the following properties :
  • $d(u.,w)=0\iff u=v$
  • $d(V, W) =0\iff V\cap W\neq \emptyset$
I have done the following:

  • $d(u, w) =0\iff |u-w|=0\iff u-w=0\iff u=w$

    Or do we have to do more steps?

    $$$$
  • $d(V, W) =0\iff \min \{d(v, w) \} =0$ this means that there exists $v$ and $w$ such that $d(v, w) =0$ and from the previous one it follows that $v=w$ which means that the intersection is non empty.

    Is that correct?

(Wondering)
 
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  • #2
Hey mathmari!

What is your function $d$ exactly? (Wondering)
 
  • #3
Klaas van Aarsen said:
What is your function $d$ exactly? (Wondering)

The following definitions are given:

  • $v$ is orthogonal to $w$, if $v\cdot w=0_{\mathbb{R}}$.
  • $|v|:=\sqrt{v\cdot v}$ is the lengthh of the vector $v$.
  • $d(v,w):=|v-w|$ is the distance between the vectors $v$ and $w$.
    $d(V,W):=\min \{d(v,w)\mid v\in V, w\in W\}$ is the distance between $V$ and $W$.
 
  • #4
mathmari said:
The following definitions are given:

  • $v$ is orthogonal to $w$, if $v\cdot w=0_{\mathbb{R}}$.
  • $|v|:=\sqrt{v\cdot v}$ is the lengthh of the vector $v$.
  • $d(v,w):=|v-w|$ is the distance between the vectors $v$ and $w$.
    $d(V,W):=\min \{d(v,w)\mid v\in V, w\in W\}$ is the distance between $V$ and $W$.
That definition of $d(V,W)$ is very suspicious because the minimum value $\min \{d(v,w)\mid v\in V, w\in W\}$ is not usually attained. For example, if you define $V, W\subset \Bbb{R}$ to be the open intervals $V=(0,1)$ and $W=(1,2)$ then $\inf \{d(v,w)\mid v\in V, w\in W\} = 0$. But $\min \{d(v,w)\mid v\in V, w\in W\}$ does not exist. You can find $v\in V$ and $w\in W$ with $d(v,w)$ arbitrarily close to $0$. But that distance can never be equal to $0$.
 
  • #5
mathmari said:
The following definitions are given:

  • $v$ is orthogonal to $w$, if $v\cdot w=0_{\mathbb{R}}$.
  • $|v|:=\sqrt{v\cdot v}$ is the lengthh of the vector $v$.
  • $d(v,w):=|v-w|$ is the distance between the vectors $v$ and $w$.
    $d(V,W):=\min \{d(v,w)\mid v\in V, w\in W\}$ is the distance between $V$ and $W$.

With that definition I believe that your proofs are all correct. (Happy)

Opalg said:
That definition of $d(V,W)$ is very suspicious because the minimum value $\min \{d(v,w)\mid v\in V, w\in W\}$ is not usually attained. For example, if you define $V, W\subset \Bbb{R}$ to be the open intervals $V=(0,1)$ and $W=(1,2)$ then $\inf \{d(v,w)\mid v\in V, w\in W\} = 0$. But $\min \{d(v,w)\mid v\in V, w\in W\}$ does not exist. You can find $v\in V$ and $w\in W$ with $d(v,w)$ arbitrarily close to $0$. But that distance can never be equal to $0$.

Suspicious or not, it merely means that the distance of such open intervals is not defined, isn't it?
It would probably be 'nicer' if they used $\inf$ instead of $\min$, but for the problem at hand it does not really matter does it?
Those $V$ and $W$ are probably intended to be linear or affine sub spaces or some such, in which case the distinction would not be relevant. (Thinking)
 
  • #6
Klaas van Aarsen said:
With that definition I believe that your proofs are all correct. (Happy)

Can we just say that $|u-w|=0\iff u-w=0$ ? Or do we have to use some more steps in between? (Wondering)
 
  • #7
mathmari said:
Can we just say that $|u-w|=0\iff u-w=0$ ? Or do we have to use some more steps in between?

Well, $|\cdot|$ is the Euclidean norm here.
Part of the definition of a norm, is that $|x|=0 \iff x=0$.
So as such we can simply use that. (Nod)
We might mention that it is because $|\cdot|$ is a norm.

To be fair, your $d$ is the Euclidean metric.
Part of the definition of a metric, is that $d(x,y)=0 \iff x=y$ (point separating).
So there was no actual need to prove it. We could have referred to the definition of a metric instead. (Nerd)
 
  • #8
@mathmari

Just a note to explain why we need to be careful. The metric (distance function, norm, whatever) you are using is Eudlidean and has all the familiar properties, such as \(\displaystyle x = 0 \implies ||x|| = 0\). That's because the metric that Klaas van Aarsen mentioned (and is also the one you are using) is positive definite, which means the norm is either 0 or real.

But there are other useful metrics. For example, in SR, we have an indefinite metric, which means that \(\displaystyle x \cdot x\) can be negative, zero, or positive. (Setting c = 1 for convenience the Minkowski metric says that the vector <0, 0, 1, 0> has a norm of -1 and that the vector <1, 0, 1, 0> has a norm of 0.)

There are all sorts of fun metrics out there. (Sun)

-Dan
 
  • #9
For the distance between two sets, it would be better to use "least upper bound" rather than "min". In R1 the distance between U= (0, 1) and V= (2, 3) is "lub(|x- y|, x in U, y in V)= 1" while "min(|x- y|, x in U. y in V)" does not exist.
 
  • #10
topsquark said:
@mathmari

Just a note to explain why we need to be careful. The metric (distance function, norm, whatever) you are using is Eudlidean and has all the familiar properties, such as \(\displaystyle x = 0 \implies ||x|| = 0\). That's because the metric that Klaas van Aarsen mentioned (and is also the one you are using) is positive definite, which means the norm is either 0 or real.

But there are other useful metrics. For example, in SR, we have an indefinite metric, which means that \(\displaystyle x \cdot x\) can be negative, zero, or positive. (Setting c = 1 for convenience the Minkowski metric says that the vector <0, 0, 1, 0> has a norm of -1 and that the vector <1, 0, 1, 0> has a norm of 0.)

There are all sorts of fun metrics out there. (Sun)

-Dan

This has been a source of confusion to me. ;)

The so called Minkowski metric does not satisfy the generally used definition of a metric.
Interestingly, the wiki article of a metric does not even mention the Minkowski metric. It does mention various other flavors, such as pseudometrics, quasimetrics, and semimetrics, but the Minkowski metric is none of those!

Moreover, the wiki article for the Minkowski metric does not make any mention that it is not a metric according to the usual definition of a metric.
Confusing indeed!

Either way, as I interpret it, the Minkowski metric is formally not a metric. Instead it is a metric tensor. (Nerd)
 
  • #11
Klaas van Aarsen said:
Either way, as I interpret it, the Minkowski metric is formally not a metric. Instead it is a metric tensor. (Nerd)
(Swearing) Jim, I'm a Physicist, not a Mathematician!

Good catch!

-Dan
 

Related to What are the properties of distance?

1. What is the definition of distance?

Distance is the numerical measurement of how far apart two points are in space. It is typically measured in units such as meters, kilometers, or miles.

2. How is distance calculated?

The distance between two points can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the other two sides. In other words, distance = √(x2-x1)2 + (y2-y1)2, where (x1,y1) and (x2,y2) are the coordinates of the two points.

3. What are the properties of distance?

The properties of distance include: it is always positive, it is symmetric (the distance from point A to point B is the same as the distance from point B to point A), and it obeys the triangle inequality (the sum of any two sides of a triangle is greater than the third side).

4. How is distance used in real life?

Distance is used in various fields such as navigation, transportation, and physics. It is also used in everyday situations, such as measuring the distance between two locations on a map, calculating the length of a road trip, or determining the distance between two objects in a room.

5. How can we prove the properties of distance?

The properties of distance can be proven using mathematical proofs and logical reasoning. For example, to prove that distance is always positive, we can assume a negative distance and show that it leads to a contradiction. To prove the triangle inequality, we can use the Pythagorean theorem and show that the sum of any two sides of a triangle is always greater than the third side.

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