NAME: 4 MT263 problems Hand in your solutions, with this coversheet, for feedback the Maths Office (McCrea 118) by 1pm on Monday 11th Feb. I will keep a record of how many questions you made a decent attempt on. I can provide you with additional feedback if you want it. Many students don't collect their feedback so if you don't want additional feedback, please tick the relevant box below. Otherwise, if you make a decent attempt of N questions, you can request additional feedback on [ { ] questions (note the use of the floor function here). Please indicate which questions you'd like feedback on in the relevant box below (if you don't indicate what feedback you want, I'll assume you don't want any). • Put a tick in this box if you don't want additional written feedback on your solutions. · I attempted questions. Please provide me with feedback on questions YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1. Which of the following sets are convex? Justify your answers. (a) { (x,y) E R2 : 1 ?x ? 2, y = 1}. Solution: The set is convex. Let a = (a1, @2), b = (b1, b2) be two points from the set. Then a2 = b2 = 1 and hence Aa2 + (1 - 1)62 = 1 for all 0 < > < 1. In addition 1 < a1, b1 < 2 and hence 1? Aa1 + (1 - A)62 ? 2. Thus Aa + (1 - A)b is in the set. (b) {(x, y) E R2 : x > 0, y > 0, y ? 1/x}. Solution: The set is not convex. Consider the points (1/2, 2) and (4, 1/4) which are in the set, but the point 1/2(1/2, 2) + 1/2(4, 1/4) = (9/4, 9/8) is not.
(c) {(x, y) ? IR2 : 0 ? x ? 1,0 ?y ?2} {(x, y) ? R2 : 0 ? x ?2,0?y?1}. Solution: Not convex. The point (1, 2) is in the first set and hence in the union and the point (2, 1) is in the second set and hence in the union. But the point 1/2(1, 2) + 1/2(2, 1) = (3/2, 3/2) is in none of the sets. 2. Which of the following statements are true? Justify your answers (i.e., give a proof of, or a counter example to, each statement). (a) The intersection of two convex sets in R" is convex. Solution: True. Take two convex sets A and B. Let a, b E AnB and 0 < > < 1. Then a, b E A and a, b & B. Since A is convex Aa + (1 - 1)b is in A and since B is convex Aa + (1 - 1)b is also in B and hence also in An B. (b) The union of two convex sets is convex. Solution: False. For example, the sets {0} and {1} of