How to Prove the Triangle Inequality Property for the Metric d?

[kingwinner]

Homework Statement

Let (X,ρ) and (Y,σ) be metric spaces.
Define a metric d on X x Y by d((x1,y1),(x2,y2))=max(ρ(x1,x2),σ(y1,y2)).
Verify that d is a metric.

Homework Equations

The Attempt at a Solution

I proved positive definiteness and symmetry, but I am not sure how to prove the "triangle inequality" property of a metric. How many cases do we need in total, and how can we prove it?

Any help is appreciated!

 

[kingwinner]

So to verify the triangle inequality, we need to prove that
max(ρ(x1,x2),σ(y1,y2))≤ max(ρ(x1,x3),σ(y1,y3)) + max(ρ(x3,x2),σ(y3,y2)) for ANY (x1,y1),(x2,y2),(x3,y3) in X x Y.

How many separate cases do we need? I have trouble counting them without missing any...Is there a systematic way to count?

Case 1: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=ρ(x1,x3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

This case is simple, the above inequality is true since ρ is a metric.


Case 2: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=σ(y1,y3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

For example, how can we prove case 2?


Any help is appreciated!

 

[Tinyboss]

Suppose that $$\rho(x_1,x_2)\ge\sigma(y_1,y_2)$$. What do you know about $$\rho(x_1,x_3)+\rho(x_3,x_2)$$? Can you infer anything about the right-hand side of your inequality based on that?

 

[kingwinner]

Case 2: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=σ(y1,y3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

Suppose that $$\rho(x_1,x_2)\ge\sigma(y_1,y_2)$$. What do you know about $$\rho(x_1,x_3)+\rho(x_3,x_2)$$? Can you infer anything about the right-hand side of your inequality based on that?

We'll have ρ(x1,x3)+ρ(x3,x2) ≥ σ(y1,y2)).

But I think for case 2, we need to prove that ρ(x1,x2)≤σ(y1,y3)+ρ(x3,x2) instead?? How can we prove it?

Thanks!

 

[kingwinner]

Still confused...please help...

 

[Tinyboss]

Okay, you need to show, for all points, that

$$\max(\rho(x_1,x_2),\sigma(y_1,y_2))\le\max(\rho(x_1,x_3),\sigma(y_1,y_3))+\max(\rho(x_3,x_2),\sigma(y_3,y_2))$$.

Suppose $$\rho(x_1,x_2)\ge\sigma(y_1,y_2)$$. Since $$\rho$$ is a metric, you know that $$\rho(x_1,x_3)+\rho(x_3,x_2)\ge\rho(x_1,x_2)$$.

So what do you know about $$\max(\rho(x_1,x_3),\square)+\max(\rho(x_3,x_2),\square)$$, regardless of what's in the squares? You know it's at least as big as $$\rho(x_1,x_2)$$.