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Q5E

Expert-verifiedFound in: Page 437

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.**

- Not compact but convex
- Compact and convex
- Not compact but convex
- Not compact and not convex
- Not compact but convex

(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.

(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.

(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.

(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.

(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.

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