Posted: February 26th, 2022

Advanced Analysis

Portfolio Problems

Portfolio Draft 1 – Due Feb 25

1. Cardinality of sets.

(a) Task 1: Explain the ideas of cardinality in your own words. (Definitions 0.3.26 – 0.3.29)

(b) Task 2: Pick one of the following and use the definition of cardinality to prove or disprove the statement.

• Problem A: Z and the set E of even natural numbers have the same cardinality. • Problem B: The intervals (3, 5) and (2,∞) have the same cardinality.

2. Supremum and Infimum of sets.

(a) Task 1: Explain the ideas of bounds, supremum, and infimum in your own words. (Definition 1.1.2 – 1.1.3)

(b) Task 2: Do one of the following.

• Problem A: Let E = {5 − 1 n

: n ∈ N}. Determine sup E and inf E and prove your answers.

• Problem B: Prove: If x < 0 and A ⊂ R is bounded below then sup(xA) = x inf A

3. Properties of the real line.

(a) Task 1: Select a few of the special properties of R we’ve talked about (Theorem 1.2.1, Archimedean Property, Q is dense in R, R is uncountable, cardinality of intervals in R, etc) and explain what you understand about them in your own words. (e.g. What is the property? Why is it interesting or useful? etc)

(b) Task 2: Use the property that Q is dense in R to prove one of the following. • Problem A: Let a ∈ R and B = {x ∈ Q : x > a}. Determine inf B and prove your

answer.

• Problem B: The set of irrational numbers is dense in R. That is, between any two real numbers x < y there is an irrational number s so that x < s < y.

4. Supremum and Infimum of functions.

(a) Task 1: Explain what it means for a function to be bounded and what the supremum and infimum of a function is. (Def 1.3.6, Figure 1.3)

(b) Task 2: Complete one of the following problems.

• Problem A: Let f : D → R and g : D → R be bounded functions. Prove that f + g : D → R is bounded and sup

x∈D (f + g) ≤ sup

x∈D f + sup

x∈D g.

• Problem B: Let h : [−3, 2] → R be given by h(x) = 20 − x3 − x2 + 7x. Prove that h is bounded and find sup

x∈[−3,2] h and inf

x∈[−3,2] h.

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Recall the tasks: For each learning objective students will…

1. Explain the concept in their own words, providing pictures and examples as appropriate to demonstrate their understanding.

2. Homework help – Write a complete and polished proof or solution, selected from a list of problems.

3. Reflect on what they learned about the topic.

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