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.

(a)

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

This set is convexas this is not closed and thus not bounded.

Hence, the set \(\left\{ {\left( {x,y} \right):y > 0} \right\}\) is not compactbut 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.