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**acevipa** Suppose that $\displaystyle A$ and $\displaystyle B$ are non-empty subsets of $\displaystyle \mathbb{R}^2.$ Define the distance $\displaystyle d(A,B)$ between $\displaystyle A$ and $\displaystyle B$ by the formula:

$\displaystyle d(A,B)=inf\{|\mathbf{a}-\mathbf{b}|:\mathbf{a}\in A, \mathbf{b}\in B\}$

1) Explain why this infimum always exists

2) Suppose that $\displaystyle A=\{(x,y) : x^2+y^2<1\} and B=\{(x,y) : x^2-y^2>9\}$. Find $\displaystyle d(A,B)$

3) Find 2 disjoint sets $\displaystyle A$ and $\displaystyle B$ such that $\displaystyle d(A,B)=0$