# Is {X, max(d,r)} or (X, min(d,r)) a Metric Space?

• ag2ie
In summary, the conversation is discussing whether the space {X, max(d, r)} is a metric space, and if the space (X, min(d, r)) is also a metric space. The first person has verified that d(x,y)=0 iff x=y and d(x, y)=d(y,x), but is unsure how to prove the triangle inequality. The second person suggests an inequality to try, but the first person points out that it may not always hold true. They both agree that (X, min(d, r)) is not a metric space.

#### ag2ie

If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?

Well, what do you think?? What have you tried?

It's easy to verify d(x,y)=0 iff x=y and d(x, y)=d(y,x),
but I don't know how to prove triangle Inequality...

Well, is the following trye

$$d(x,z)\leq \max\{d(x,y),r(x,y)\}+\max\{d(y,z),r(y,z)\}$$

?

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yes..but if r(x,z) is greater than d(x,z), and r(y,z) is smaller than d(y,z), then this inequality is not necessary true...right?

Sorry, I made a typo, check the post again.

I see...if r(x,y) is greater than d( x,y), then d(x,z)≤max{d(x,y),r(x,y)}+max{d(y,z),r(y,z)} is also true...

Thanks ...and I think (X, min(d, r)) is not a metric space..right?

ag2ie said:
Thanks ...and I think (X, min(d, r)) is not a metric space..right?

right.

## 1. Is a Metric Space always a set of numbers?

No, a Metric Space can be defined as a set of any objects as long as there is a distance function that satisfies the properties of a metric.

## 2. What are the properties that a distance function in a Metric Space must satisfy?

The distance function in a Metric Space must satisfy the properties of non-negativity, symmetry, and the triangle inequality.

## 3. Can a Metric Space have more than one distance function?

Yes, a Metric Space can have multiple distance functions as long as they all satisfy the properties of a metric.

## 4. How is the distance between two points in a Metric Space calculated?

The distance between two points in a Metric Space is calculated using the distance function, which takes the two points as input and returns a value that represents the distance between them.

## 5. Can a Metric Space have an infinite number of points?

Yes, a Metric Space can have an infinite number of points as long as the distance function is well-defined for all pairs of points.