quiz1 - SOLUTIONS FOR QUIZ #1. 1(a) Prove that the set S =...

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SOLUTIONS FOR QUIZ #1. 1(a) Prove that the set S = { ( x,y ) : 1 < x 2 + y 2 } in R 2 is open. You have to give a rigorous proof based on the de nition of an open set. We have to show that x S, r > 0 such that B ( r, x ) S. Let r = || x || - 1 . Then y B ( r, x ) = {|| x - y || < r } we have || y || ≥ || x || - || x - y || > || x || - r = || x || - || x || + 1 = 1 (by exercise #6, sec.1.1, a corollary to triangle inequality || a || - || b || ≤ || a - b || ) so || y || > 1 and y S meaning that B ( r, x ) S. (b) Give an example of a set S such that the interior of S is unequal to the interior of the closure of S. Justify your answer. Let S = { ( x,y ) : 0 < x 2 + y 2 < 1 } ⇒ S = S int ¯ S = { ( x,y ) : x 2 + y 2 1 } ⇒ ( ¯ S ) int = { ( x,y ) : x 2 + y 2 < 1 } 6 = S int (other examples possible) 2(a) Show directly from the de nition that the set S = { ( x,y ) R 2 : xy = 1 } is disconnected. That is produce a disconnection (in a set notation) for S. S
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quiz1 - SOLUTIONS FOR QUIZ #1. 1(a) Prove that the set S =...

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