Prove that the metric area is defined as P &lt; x1, y1 &gt; and Q &lt; x2, y2 &gt;. 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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Prove that the metric area is defined as P &amp;lt; x1, y1 &amp;gt; and Q &amp;lt; x2, y2 &amp;gt;. 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.

Latisha Oneil · Accepted Answer

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

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.

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