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

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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

SLIDE 1

Lecture Problems Week #1b

Parker Glynn-Adey and Tyler Holden May 12, 2016

1 / 21

SLIDE 2

Question

Can a subset of Rn be neither open nor closed?

1 Yes 2 No

2 / 21

SLIDE 3

Question

Can a subset of Rn be both open and closed?

1 Yes 2 No

3 / 21

SLIDE 4

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

SLIDE 5

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

SLIDE 6

Question

Let S = {(x, y) ∈ R2 : 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

SLIDE 7

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

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SLIDE 8

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

SLIDE 9

Question

Let f : R2 → R be defined f (x, y) = x2 + y2. 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

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SLIDE 10

Question

What is (0, 1)2 ⊆ R2? 1) 2) 3) 4)

10 / 21

SLIDE 11

Question

What is ∂[0, 1] in R?

1 (0, 1) 2 {0, 1} 3 ∅ 4 [0, 1]

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SLIDE 12

Question

Let S = {(x, sin(x)) : x ∈ R} ⊆ R2. Is S closed and bounded?

1 Bounded and not closed 2 Both 3 Closed and not bounded 4 Neither

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SLIDE 13

Question

Let S =      1 2 3   ·   x y z   = 0   . Is S closed and bounded?

1 Bounded and not closed 2 Both 3 Neither 4 Closed and not bounded

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SLIDE 14

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

SLIDE 15

Question

What is ∂ 1

n

n∈N in R?

1 (0, 1) 2

1

n

n∈N

3 {0} ∪

1

n

n∈N

4 {0} 5 ∅

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SLIDE 16

Question

How many boundary points does R have?

1 Two 2 Infinitely many 3 One 4 None

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SLIDE 17

Question

What is ∂Q as a subset of R?

1 ∅ 2 Q 3 R 4 R \ Q

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SLIDE 18

Question

Suppose S ⊂ Rn. What is Int(S) ∩ S?

1 Int(S) 2 S 3 ∅ 4 S

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SLIDE 19

Question

Suppose S ⊂ Rn. What is (∂S) ∩ S?

1 ∅ 2 S 3 S 4 ∂S

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SLIDE 20

Question

Let S = Q. What is Int(S)?

1 R 2 Q 3 R \ Q 4 ∅

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