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Chapter 8: Q4E (page 437)

Question: In Exercise 4, determine whether each set is open or closed or neither open nor closed.

4. a. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} = {\bf{1}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} > {\bf{1}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} \le {\bf{1}}\,\,\,and\,\,y > {\bf{0}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):y \ge {x^{\bf{2}}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):y < {x^{\bf{2}}}} \right\}\)

Short Answer

Expert verified
  1. Closed
  2. Open
  3. Neither open nor closed
  4. Closed
  5. Open

Step by step solution

01

Use the fact of a circle equation

(a)

Note that the given set is the set of all points on a circle centred at the origin with radius 1.

Hence it contains all its boundary points.

This implies \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} = 1} \right\}\) is closed.

02

Use the fact of complement

(b)

It is known that the complement of a closed set is an open set.

From part (a), the complement of \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} = 1} \right\}\) is open. That is, \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} > 1\,\,{\rm{and}}\,\,{x^2} + {y^2} < 1} \right\}\) open.

This implies \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} > 1\,} \right\}\)and \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} < 1\,} \right\}\) are open.

03

Use fact of a circle equation

(c)

The given set contains all the points of the upper half of the circle centred at the origin with radius one and above the x-axis. Hence it does not include all its boundary points, and it is not open as well.

This implies \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1\,\,{\rm{and}}\,y > 0} \right\}\) is neither open nor closed.

04

Use the fact of a closed set

(d)

The given set contains all its boundary points.

Hence, \(\left\{ {\left( {x,y} \right):y \ge {x^2}} \right\}\) is closed.

05

Use the fact of complement

(e)

It is known that the complement of a closed set is open.

From part (d), the complement of \(\left\{ {\left( {x,y} \right):y \ge {x^2}} \right\}\) is open. That is, \(\left\{ {\left( {x,y} \right):y < {x^2}} \right\}\) open.

Thus, the set is open.

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Most popular questions from this chapter

In Exercises 21–24, a, b, and c are noncollinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \) and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

21. Show that the area of\(\Delta {\bf{abc}}\)is\(det\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]/2\).

[Hint:Consult Sections 3.2 and 3.3, including the Exercises.]

Question: 24. Repeat Exercise 23 for \({v_1} = \left( \begin{array}{l}1\\2\end{array} \right)\), \({v_2} = \left( \begin{array}{l}5\\1\end{array} \right)\), \({v_3} = \left( \begin{array}{l}4\\4\end{array} \right)\) and \(p = \left( \begin{array}{l}2\\3\end{array} \right)\).

Let \({\bf{x}}\left( t \right)\) be a B-spline in Exercise 2, with control points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\) , \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\) and determine how the derivatives \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points. Give geometric descriptions of the directions of these tangent vectors. Explore what happens when both \({\bf{x}}'\left( 0 \right)\)and \({\bf{x}}'\left( 1 \right)\)equal 0. Justify your assertions.

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_2}\) as the origin of the coordinate system.]

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{5}}\\{\bf{0}}&{\bf{2}}&{\bf{4}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\). (Hint: How is H is related to Nul B?see section 6.1.)

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

10.a. If \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely dependent set in \({\mathbb{R}^n}\), then the set \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) of homogeneous forms may be linearly independent.

b. If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) and \({{\mathop{\rm v}\nolimits} _4}\) are in \({\mathbb{R}^3}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _4} - {{\mathop{\rm v}\nolimits} _1}} \right\}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) is affinely independent.

c. Given \(S = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}} \right\}\) in \({\mathbb{R}^n}\), each \({\bf{p}}\) in\({\mathop{\rm aff}\nolimits} S\) has a unique representation as an affine combination of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}\).

d. When color information is specified at each vertex \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) of a triangle in \({\mathbb{R}^3}\), then the color may be interpolated at a point p in \({\mathop{\rm aff}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) using the barycentric coordinates of p.

e. If T is a triangle in \({\mathbb{R}^2}\) and if a point p is on edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.
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