Lecture Problems Week #1b Parker Glynn-Adey and Tyler Holden May - PowerPoint PPT Presentation

Lecture Problems Week #1b Parker Glynn-Adey and Tyler Holden May 12, 2016 1 / 21

Question Can a subset of R n be neither open nor closed? 1 Yes 2 No 2 / 21

Question Can a subset of R n be both open and closed? 1 Yes 2 No 3 / 21

Question Let S = { 1 , 2 , 3 } ⊂ R . Is S open, closed, both, or neither? 1 Neither 2 Open and not closed 3 Both 4 Closed and not open 4 / 21

Question Let S = { 1 } ∪ (2 , 3) ⊂ R . Is S open, closed, both, or neither? 1 Open and not closed 2 Closed and not open 3 Both 4 Neither 5 / 21

Question Let S = { ( x , y ) ∈ R 2 : max {| x | , | y |} < 1 } . Is S open, closed, both, or neither? 1 Open and not closed 2 Closed and not open 3 Neither 4 Both 6 / 21

Question Let S = ∅ ⊂ R . Is S open, closed, both, or neither? 1 Open and not closed 2 Both 3 Neither 4 Closed and not open 7 / 21

Question Let S = R . Is S open, closed, both, or neither? 1 Closed and not open 2 Neither 3 Open and not closed 4 Both 8 / 21

Question Let f : R 2 → R be defined f ( x , y ) = x 2 + y 2 . Let S = f − 1 (( −∞ , 1)) . Is S open, closed, both, or neither? 1 Open and not closed 2 Both 3 Closed and not open 4 Neither 9 / 21

Question What is (0 , 1) 2 ⊆ R 2 ? 1) 2) 3) 4) 10 / 21

Question What is ∂ [0 , 1] in R ? 1 (0 , 1) 2 { 0 , 1 } 3 ∅ 4 [0 , 1] 11 / 21

Question Let S = { ( x , sin( x )) : x ∈ R } ⊆ R 2 . Is S closed and bounded? 1 Bounded and not closed 2 Both 3 Closed and not bounded 4 Neither 12 / 21

Question       1 x    ·  = 0 Let S = 2 y  . Is S closed and bounded?   3  z 1 Bounded and not closed 2 Both 3 Neither 4 Closed and not bounded 13 / 21

Question Let S = { ( x , y , z ) : x + y + z = 1 , x , y , z ≥ 0 } . Is S closed and bounded? 1 Bounded and not closed 2 Closed and not bounded 3 Both 4 Neither 14 / 21

Question � 1 � What is ∂ n ∈ N in R ? n 1 (0 , 1) � 1 � 2 n ∈ N n � 1 � 3 { 0 } ∪ n ∈ N n 4 { 0 } 5 ∅ 15 / 21

Question How many boundary points does R have? 1 Two 2 Infinitely many 3 One 4 None 16 / 21

Question What is ∂ Q as a subset of R ? 1 ∅ 2 Q 3 R 4 R \ Q 17 / 21

Question Suppose S ⊂ R n . What is Int( S ) ∩ S? 1 Int( S ) 2 S 3 ∅ 4 S 18 / 21

Question Suppose S ⊂ R n . What is ( ∂ S ) ∩ S? 1 ∅ 2 S 3 S 4 ∂ S 19 / 21

Question Let S = Q . What is Int( S ) ? 1 R 2 Q 3 R \ Q 4 ∅ 20 / 21

Question Let S = { 0 } . What is Int( S ) . 1 ∅ 2 { 0 } 3 R 21 / 21