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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

Question 1: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\3\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right)\) in \({\mathbb{R}^2}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a.\(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 3{x_1} + {x_2}\)

Question: In Exercise 8, let Hbe the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{2}}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{7}}\\{ - {\bf{4}}}\\{\bf{4}}\end{array}} \right)\)

Question 4: Repeat Exercise 2 where \(m\) is the minimum value of \(f\) on \(S\) instead of the maximum value.

Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).

Suppose \({\bf{y}}\) is orthogonal to \({\bf{u}}\) and \({\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to every \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\). (Hint: An arbitrary \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\) has the form \({\bf{w}} = {c_1}{\bf{u}} + {c_2}{\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to such a vector \({\bf{w}}\).)
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