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Q5E

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Found in: Page 437

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Question: In Exercise 5, determine whether or not each set is compact and whether or not it is convex.5. Use the sets from Exercise 3.

1. Not compact but convex
2. Compact and convex
3. Not compact but convex
4. Not compact and not convex
5. Not compact but convex
See the step by step solution

## Step 1: Use Exercise 3(a)

(a)

Form Exercise 3(a), it is obtained that $$\left\{ {\left( {x,y} \right):y > 0} \right\}$$ is open.

This set is convex as this is not closed and thus not bounded.

Hence, the set $$\left\{ {\left( {x,y} \right):y > 0} \right\}$$ is not compact but convex.

## Step 2: Use Exercise 3(b)

(b)

Form Exercise 3(b), it is obtained that the set $$\left\{ {\left( {x,y} \right):x = 2\,\,{\rm{and}}\,\,1 \le y \le 3} \right\}$$ is closed.

This set is convex as it is closed and bounded.

Hence, the set $$\left\{ {\left( {x,y} \right):x = 2\,\,{\rm{and}}\,\,1 \le y \le 3} \right\}$$ is compact and convex.

## Step 3: Use Exercise 3(c)

(c)

Form Exercise 3(c), it is observed that the set $$\left\{ {\left( {x,y} \right):x = 2\,\,{\rm{and}}\,\,1 < y < 3} \right\}$$ is neither open nor closed.

This set is convex and bounded.

Hence, $$\left\{ {\left( {x,y} \right):x = 2\,\,{\rm{and}}\,\,1 < y < 3} \right\}$$not compact but convex.

## Step 4: Use Exercise 3(d)

(d)

Form Exercise 3(d), it is observed that the set $$\left\{ {\left( {x,y} \right):xy = 1\,\,{\rm{and}}\,\,x > 0} \right\}$$ is closed.

This set is not convex and not bounded.

Hence, $$\left\{ {\left( {x,y} \right):y > 0} \right\}$$ not compact and not convex.

## Step 5: Use Exercise 3(e)

(e)

Form Exercise 3(e), it is observed that the set $$\left\{ {\left( {x,y} \right):xy \ge 1\,\,{\rm{and}}\,\,x > 0} \right\}$$ is closed.

This set is convex and not bounded.

Hence, $$\left\{ {\left( {x,y} \right):y > 0} \right\}$$ is not compact but convex.