How Does the Strong Triangle Inequality Justify Equal p-adic Balls?

In summary: No, that's not what I meant. The strong triangle inequality says that if r is an element of Be(s), then Be(s)=Be(r). This means that any two balls of radius e that overlap are also equal. This is reasonable because they are all centered at the same point.
  • #1
Ed Quanta
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Let e>0 and let Be(r),Be(s) denote the open balls of radius e centered at r,s with respect to the p-adic metric. Prove that if r is an element of Be(s), then Be(s)=Be(r).

Can someone show me how to use the strong triangle inequality to do this?
 
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  • #2
Is this what you mean by strong triangle inequality?

d(x,z) <= max{d(x,y), d(y,z)} for all x,y,z in whatever space.

If so, suppose that x is an element of B_e(r). That is, d(x,r) < e. We are given that d(r,s) < e. Then use the triangle inequality to show that d(x,s) < e also. So B_e(r) is contained in B_e(s). Use a similar argument to show B_e(s) is contained in B_e(r).
 
  • #3
Yes, that was what I meant by the strong triangle inequality. Why must I use this form of the triangle inequality as opposed to the more general form of the triangle inequality?

And since d(x,s)<=max(d(x,r),d(r,s))

We know d(x,r)<e and d(r,s)<e, so wouldn't this just imply

d(x,s)<2e?
 
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  • #4
Ed Quanta said:
Why must I use this form of the triangle inequality as opposed to the more general form of the triangle inequality?
Because the regular triangle inequality only gets you d(x,s) < 2e, like you showed. This stronger version says that d(x,s) is less than or equal to the larger of these other two "distances" d(x,r) and d(r,s), both of which are strictly less than e. That is, d(x,s) < e. You need this to show that x is also in B_e(s).
 
  • #5
And, as a corollary, i you didnt' need the strong one then yo'uve said that all balls of radius e that over lap are equal, and that doesn't seem at all reasonable does it?
 

Related to How Does the Strong Triangle Inequality Justify Equal p-adic Balls?

1. What is a metric space?

A metric space is a mathematical concept that defines the distance between objects in a given space. It is a set of points where the distance between any two points is defined by a function called a metric.

2. What are the properties of a metric space?

The properties of a metric space include non-negativity (the distance between any two points is always positive or zero), symmetry (the distance from point A to point B is equal to the distance from point B to point A), and the triangle inequality (the distance from point A to point C is always less than or equal to the sum of the distances from point A to point B and from point B to point C).

3. How is a metric space different from a normed space?

A metric space and a normed space are both mathematical concepts that define the distance between objects in a given space. However, a normed space also includes the concept of direction and magnitude, while a metric space only defines the distance between points.

4. What are some examples of metric spaces?

Some examples of metric spaces include Euclidean space (the familiar 3-dimensional space we live in), Manhattan space (where distance is measured by the sum of the absolute differences in coordinates), and discrete space (where the distance between two points is either 0 or 1).

5. How are metric spaces used in real-world applications?

Metric spaces are used in a variety of real-world applications, such as in mathematics, physics, computer science, and data analysis. They are particularly useful in fields such as data mining and machine learning, where distance measures are used to identify patterns and make predictions.

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