What is Meant by By Symmetry in the Reverse Triangle Inequality Proof?

[MaxManus]

Homework Statement

I'm reading the proof for the reverse triangle inequality, but I don't understand what is meant by "by symmetry"

Homework Equations

The Attempt at a Solution

(X,d) is a metric space
prove:
|d(x,y) - d(x,z)| <= d(z,y)

The triangle inequality
d(x,y) <= d(x,z) + d(z,y)
d(x,y) - d(x,z) <= d(z,y)

By symmetry
d(x,z) - d(x,y) <= d(y,z) = d(z,y)

So:
|d(x,y) - d(x,z)| <= d(z,y)

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How did they get the "by symmetry d(x,z) - d(x,y) <= d(y,z) = d(z,y)" part?
Isn't symmetry just that d(x,y) = d(y,x)

 

[angst18]

I believe the symmetry reference is due to the fact that for a metric space $d(x,y) = d(y,x)$.

 

[MaxManus]

Yes, but how does that take you to: "d(x,z) - d(x,y) <= d(y,z) = d(z,y)"?

 

[micromass]

Hi MaxManus!

By symmetry is a standard mathematical phrase and has nothing to do with symmetry here. It just means that the property is analogous to something you have done before.

Homework Statement

I'm reading the proof for the reverse triangle inequality, but I don't understand what is meant by "by symmetry"

Homework Equations

The Attempt at a Solution

(X,d) is a metric space
prove:
|d(x,y) - d(x,z)| <= d(z,y)

The triangle inequality
d(x,y) <= d(x,z) + d(z,y)
d(x,y) - d(x,z) <= d(z,y)

By symmetry
d(x,z) - d(x,y) <= d(y,z) = d(z,y)

Here, they mean d(x,z)<=d(x,y)+d(y,z), thus d(x,z)-d(x,y)<= d(y,z). This is quite the same thing as you done before...

 

[MaxManus]

Thanks!
angst18 I tried to solve your problem, but it was over my level.

 

[angst18]

Thanks, it seems I'm doomed :/