Question

# 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.

Transformation properties
Prove that the metric area is defined as $$\displaystyle{P}\ {<}\ {x}_{{{1}}},\ {y}_{{{1}}}\ {>}\ {\quad\text{and}\quad}\ {Q}\ {<}\ {x}_{{{2}}},\ {y}_{{{2}}}\ {>}$$. 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.

2021-02-04
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:
$$\displaystyle{I}{)}\ {d}\ \text{must be a real valued, finite and nonnegative.}$$
$$\displaystyle{I}{I}{)}\ {d}\ {\left({x},\ {y}\right)}={0}\ \text{if and only if}\ {x}={y}$$
$$\displaystyle{I}{I}{I}{)}\ {d}\ {\left({x},\ {y}\right)}={d}\ {\left({y},\ {x}\right)}$$
$$\displaystyle{I}{V}{)}\ {d}\ {\left({x},\ {y}\right)}\ \leq\ {d}\ {\left({x},\ {z}\right)}\ +\ {d}\ {\left({z},\ {y}\right)}$$
Step 2
Here, the metric should be defined as $$\displaystyle{d}\ {\left({x},\ {y}\right)}$$ must be the distance between the point x and y.
$$\displaystyle{d}\ {\left({P},\ {Q}\right)}=\sqrt{{{\left({x}_{{{1}}}\ -\ {x}_{{{2}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{1}}}\ -\ {y}_{{{2}}}\right)}^{{{2}}}}}$$
$$\displaystyle\text{Try to satisfy}\ {d}\ {\left({P},\ {Q}\right)}\ \leq\ {d}\ {\left({P},\ {R}\right)}\ +\ {d}\ {\left({R},\ {Q}\right)}\ \text{where}$$
$$\displaystyle{d}\ {\left({P},\ {R}\right)}=\sqrt{{{\left({x}_{{{1}}}\ -\ {x}_{{{3}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{1}}}\ -\ {y}_{{{3}}}\right)}^{{{2}}}}}\ \text{and}\ {d}\ {\left({R},\ {Q}\right)}=\sqrt{{{\left({x}_{{{3}}}\ -\ {x}_{{{2}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{3}}}\ -\ {y}_{{{2}}}\right)}^{{{2}}}}}$$