- #1
MaxManus
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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)