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.

\(\displaystyle{d}\ {\left({P},\ {Q}\right)}=\sqrt{{{\left({x}_{{{1}}}\ -\ {x}_{{{2}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{1}}}\ -\ {y}_{{{2}}}\right)}^{{{2}}}}}\)

\(\displaystyle\text{Try to satisfy}\ {d}\ {\left({P},\ {Q}\right)}\ \leq\ {d}\ {\left({P},\ {R}\right)}\ +\ {d}\ {\left({R},\ {Q}\right)}\ \text{where}\)

\(\displaystyle{d}\ {\left({P},\ {R}\right)}=\sqrt{{{\left({x}_{{{1}}}\ -\ {x}_{{{3}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{1}}}\ -\ {y}_{{{3}}}\right)}^{{{2}}}}}\ \text{and}\ {d}\ {\left({R},\ {Q}\right)}=\sqrt{{{\left({x}_{{{3}}}\ -\ {x}_{{{2}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{3}}}\ -\ {y}_{{{2}}}\right)}^{{{2}}}}}\)

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.

\(\displaystyle{d}\ {\left({P},\ {Q}\right)}=\sqrt{{{\left({x}_{{{1}}}\ -\ {x}_{{{2}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{1}}}\ -\ {y}_{{{2}}}\right)}^{{{2}}}}}\)

\(\displaystyle\text{Try to satisfy}\ {d}\ {\left({P},\ {Q}\right)}\ \leq\ {d}\ {\left({P},\ {R}\right)}\ +\ {d}\ {\left({R},\ {Q}\right)}\ \text{where}\)

\(\displaystyle{d}\ {\left({P},\ {R}\right)}=\sqrt{{{\left({x}_{{{1}}}\ -\ {x}_{{{3}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{1}}}\ -\ {y}_{{{3}}}\right)}^{{{2}}}}}\ \text{and}\ {d}\ {\left({R},\ {Q}\right)}=\sqrt{{{\left({x}_{{{3}}}\ -\ {x}_{{{2}}}\right)}^{{{2}}}\ +\ {\left({y}_{{{3}}}\ -\ {y}_{{{2}}}\right)}^{{{2}}}}}\)