Exercise · ZW-OPEN-BALL-003 · ★☆☆ Basic
In Euclidean \(\mathbb{R}^2\), which point lies in \(B((0,0),1)\)?
Choose one option.
Show answer
- A. \((0.5,\,0.5)\)
Explanation
\(0.5^2+0.5^2=0.5<1\). The points \((1,0)\) and \((0.6,0.8)\) are at distance exactly \(1\), and \((0.8,0.7)\) at \(\sqrt{1.13}>1\).
Review the concept: Open ball
The open ball \(B(x,r)\) with center \(x\) and radius \(r>0\) in a metric space \((X,d)\) is the set of points of \(X\) whose distance to \(x\) is less than \(r\).
\[B(x,r)=\{\,y\in X : d(x,y)
More practice problems: open ball
- ZW-OPEN-BALL-001 In ℝ with d(x,y)=|x-y|, which interval is the open ball B(2,3)?
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- ZW-OPEN-BALL-004 With the max distance d_∞ on ℝ^2, the ball B((0,0),2) is a square. What is its area?
- ZW-OPEN-BALL-005 With the taxicab distance d_1 on ℝ^2, the ball B((0,0),1) is a rhombus with vertices (pm1,0) and (0…
- ZW-OPEN-BALL-006 In a space with the discrete metric, what is the ball B(a,1)?
- ZW-OPEN-BALL-007 In the usual ℝ, what is the intersection B(0,2)∩ B(3,2)?
- ZW-OPEN-BALL-008 How many points (a,b) with integer coordinates lie in the Euclidean ball B((0,0),3)?
- ZW-OPEN-BALL-009 Let y∈ B(a,r) and s=r-d(a,y). In ℝ with a=0, r=5 and y=3, what is s, the radius ensuring B(y,s)⊆ B(…