Let $(X,d)$ be a metric space. Then the function $d_H:\mathcal P(X)\times\mathcal P(X)\to\mathcal [0,\infty]$ defined by $$d_H(A,B)= \max \{ \sup\limits_{x\in A} \inf\limits_{y\in B} d(x,y), \ \sup\limits_{y\in B} \inf\limits_{x\in A} d(x,y) \}$$ for all $A,B\subset X$. This $d_H$ is called Hausdorff distance.
Now I want to check $d_H (A,B)$ for some particular case.
Consider the metric space $(\mathbb R^2,d)$ with euclidean metric $d$. Let $B=\{(x,y):x^2+y^2\leq16 \}$ and $A=\{(x,y):(x+2)^2+y^2\leq1 \}$ two subsets of $\mathbb R^2$. Here $A\subset B$. Then I have checked that first part of the definition of $d_H$ is $0$ and second part is $5$. Thus $d_H (A,B)$ is equal to $5$, being the maximum of $0,5$. Am I wrong?
I am confused because of this figure in Wikipedia.
My question is if $P,Q$ is any two subset of $(X,d)$ with $P\subset Q$, then is it true that $\sup\limits_{x\in P} \inf\limits_{y\in Q} d(x,y)=0$?
First question: I think your answer is correct. $d_H(A,B)=5$.
Second question. YES, again. If $x \in P$ then $x \in Q$ so $\inf_{y \in Q} d(x,y)\leq d(x,x)=0$. Take sup over $x$.