Prove that the metric area is defined as P < x_{1}, y_{1} > and Q < x_{2}, y_{2} >. If the proof of examples says that the first properties (positive definiteness and symmetry) are trivial. Prove the versatility of properties for a given space.

Wotzdorfg 2021-02-03 Answered
Prove that the metric area is defined as P < x1, y1 > and Q < x2, y2 >. If the proof of examples says that the first properties (positive definiteness and symmetry) are trivial. Prove the versatility of properties for a given space.
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Expert Answer

Latisha Oneil
Answered 2021-02-04 Author has 100 answers
Step 1
Firstly, the metric should be defined. The properties must be remembered always while working on metric spaces.
The metric properties are, if d is a metric, given as:
I) d must be a real valued, finite and nonnegative.
II) d (x, y)=0 if and only if x=y
III) d (x, y)=d (y, x)
IV) d (x, y)  d (x, z) + d (z, y)
Step 2
Here, the metric should be defined as d (x, y) must be the distance between the point x and y.
d (P, Q)=(x1  x2)2 + (y1  y2)2
Try to satisfy d (P, Q)  d (P, R) + d (R, Q) where
d (P, R)=(x1  x3)2 + (y1  y3)2 and d (R, Q)=(x3  x2)2 + (y3  y2)2
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