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Unformatted text preview: MAT 321 Exam II Sample Answers Professor Sormani 1. (20 points) Let A = [ 2 , 1] [ 5 , 3). Let B = B (3 , 4) (5). Let C = { ( x, y ) : x 2 + y 2 100 } and let D = { ( x, y ) : y x 5 } . b) Write down the closures of each of these sets: Cl ( A ) = [ 2 , 1] [ 5 , 3] Cl ( B ) = { ( x, y ) : ( x 3) 2 + ( y 4) 2 5 2 } Cl ( C ) = C Cl ( D ) = D c) Write down the interiors of each of these sets: int ( A ) = ( 2 , 1) ( 5 , 3) int ( B ) = B int ( C ) = { ( x, y ) : x 2 + y 2 &gt; 100 } int ( D ) = { ( x, y ) : y &lt; x 5 } d) The boundary of a set, X, is defined to be ( X ) = Cl ( X ) \ int ( X ) . Write down the boundaries of each of these sets: ( A ) = ([ 2 , 1] { 5 , 3 } ) ( { 2 , 1 } [ 5 , 3]) which is the square made by first drawing the horizontal segments and then drawing the vertical ones. ( B ) = { ( x, y ) : ( x 3) 2 + ( y 4) 2 = 5 2 } ( C ) = { ( x, y ) : x 2 + y 2 = 100 } ( D ) = { ( x, y ) : y = x 5 } e) Suppose U is an open set lying in A, is U int ( A )? Why? Yes, because the interior is the largest open set in A or because any point in U has a ball around it which is in U and thus is also in A. f) Suppose (a,b) is an accumulation point of D, is b=a5? No, it could also be in D so b a 5 . g) Is (1 , 3) A ? No, (1 , 3) is not even in the closure because it can be surrounded by a ball which doesnt intersect with A , there is no sequence in A approaching (1 , 3) . 1 2. (20 pts) TRUE OR FALSE WITHOUT JUSTIFICATION a) If x int ( A ) then there is a radius r x &gt; 0 such that B x ( r x ) A . TRUE b) If K is a compact set in C ([0 , 1]) then K is closed and bounded. TRUE c) If K C ([0 , 1]) is closed and bounded then K is compact. FALSE d) If x Cl ( A ) then there is a sequence of points x k A such that lim...
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 Fall '08
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