Lecture Problems Week #1b

Parker Glynn-Adey and Tyler Holden

May 12, 2016

1 / 21

QuestionCan a subset of Rn be neither open nor closed?

1 Yes

2 No

2 / 21

QuestionCan a subset of Rn be both open and closed?

1 Yes

2 No

3 / 21

QuestionLet 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

QuestionLet 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

QuestionLet 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

QuestionLet 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

QuestionLet 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

QuestionLet 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

9 / 21

QuestionWhat is (0, 1)2 ⊆ R2?

1) 2)

3) 4)

10 / 21

QuestionWhat is ∂[0, 1] in R?

1 (0, 1)

2 {0, 1}3 ∅4 [0, 1]

11 / 21

QuestionLet 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

12 / 21

Question

Let S =

1

23

·xyz

= 0

. Is S closed and bounded?

1 Bounded and not closed

2 Both

3 Neither

4 Closed and not bounded

13 / 21

QuestionLet 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

QuestionWhat is ∂

{1n

}n∈N in R?

1 (0, 1)

2{1n

}n∈N

3 {0} ∪{1n

}n∈N

4 {0}5 ∅

15 / 21

QuestionHow many boundary points does R have?

1 Two

2 Infinitely many

3 One

4 None

16 / 21

QuestionWhat is ∂Q as a subset of R?

1 ∅2 Q3 R4 R \Q

17 / 21

QuestionSuppose S ⊂ Rn. What is Int(S) ∩ S?

1 Int(S)

2 S

3 ∅4 S

18 / 21

QuestionSuppose S ⊂ Rn. What is (∂S) ∩ S?

1 ∅2 S

3 S

4 ∂S

19 / 21

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

1 R2 Q3 R \Q4 ∅

20 / 21

QuestionLet S = {0}. What is Int(S).

1 ∅2 {0}3 R

21 / 21