Question

Give an example of two metric on R² which are equivalent ,substantiate your claim

Answer 1

Solution:

Consider two metric spaces on R²;

1.Euclidean metric d_e

$d_e(\underline{x},\underline{y})=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$

2. The max-metric d_m

$d_m(\underline{x},\underline{y})=$max{ |x₁-y₁|,|x₂-y₂|}

We have;

$d_m(\underline{x},\underline{y})$=max{|x₁-y₁|,|x₂-y₂|}$\leq$ $\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ =$d_e(\underline{x},\underline{y})$

Also;

$d_e(\underline{x},\underline{y})=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}=\sqrt{(|x_1-y_1|)^2+(|x_2-y_2|)^2}$

$d_e(\underline{x},\underline{y})\leq\sqrt{d_m^2(\underline{x},\underline{y})+d_m^2(\underline{x},\underline{y})}$ =$\sqrt{2}d_m(\underline{x},\underline{y})$

Hence,

$d_m(\underline{x},\underline{y})\leq d_e(\underline{x},\underline{y})\leq \sqrt{2}d_m(\underline{x},\underline{y})$

for all x,y$\epsilon$ X.

Hence, metrics d_m and d_e are equivalent.