Buy Advanced Analysis Assignment

Buy Advanced Analysis Assignment
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 a}. Determine inf B and prove your
answer.
Buy Advanced Analysis Assignment
• 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.
1
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. 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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