Math 330 - Fall 2015

Homework

Homework Problems
 

Problems to hand in

Problems to present in class
 
Problem Set 10, Monday 12/14/15 (complete list)
  1. Prove Thm. 12.13
  2. Do Project 13.15. Prove that it is a bijection.
  3. Prove Thm. 13.28
No class presentation for this problem set
Problem Set 9, Monday 11/30/15 (complete list)
  1. Do Project 11.14
  2. Let s ∈ ℝ+. Prove that there is a unique r ∈ ℝ+ such that r2=s. Hint: look at the proof of Thm. 10.25. Note that this is a repackaging of 10.26 and 10.27 together, so do not quote them as part of your proof.
  3. Write down the details of the proofs that the sum of a rational number and an irrational number is irrational, and that the product of a non-zero rational number and an irrational number is irrational.
  4. Use Prop. 11.25 and/or its converse to find a closed form for the recurrence:
    a1 = 1,
    a2 = 1,
    an+1 = 3an - an-1.
    What is the relation between this sequence and the Fibonacci numbers sequence?
Monday 12/14/15 (complete list)
  1. Do Project 11.14
  2. Let s ∈ ℝ+. Prove that there is a unique r ∈ ℝ+ such that r2=s. Hint: look at the proof of Thm. 10.25. Note that this is a repackaging of 10.26 and 10.27 together, so do not quote them as part of your proof.
  3. Write down the details of the proofs that the sum of a rational number and an irrational number is irrational, and that the product of a non-zero rational number and an irrational number is irrational.
  4. Use Prop. 11.25 and/or its converse to find a closed form for the recurrence:
    a1 = 1,
    a2 = 1,
    an+1 = 3an - an-1.
    What is the relation between this sequence and the Fibonacci numbers sequence?
Problem Set 8, Monday 11/16/15 (complete list)
  1. Prove Prop. 9.18. Hint: use induction on k ≥ 0; then extend to negative k.
  2. Prove Prop. 10.10.iii
  3. Prove Prop. 10.17.
  4. Prove Prop. 10.23.iii
Monday 12/07/15 (complete list)
  1. Prove Prop. 9.18. Hint: use induction on k ≥ 0; then extend to negative k.
  2. Prove Prop. 10.10.iii
  3. Prove Prop. 10.17.
  4. Prove Prop. 10.23.iii

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