Description

General instructions
• Your problem sets are graded on both correctness and clarity of communication. Solutions that are technically correct but poorly written will not receive full marks. Please read over your solutions carefully before submitting them.
• Solutions must be typeset electronically, and submitted as a PDF with the correct filename. Handwritten submissions will receive a grade of ZERO.
The required filename for this problem set is problem set3.pdf.
• Problem sets must be submitted online through MarkUs. If you haven’t used MarkUs before, give yourself plenty of time to figure it out, and ask for help if you need it! If you are working with a partner, you must form a group on MarkUs, and make one submission per group. “I didn’t know how to use MarkUs” is not a valid excuse for submitting late work.
Additional instructions
• For each proof by induction, clearly define the predicate (P(n)) that is relevant for your induction proof, so that you’re proving a statement with structure ∀n ∈ N, P(n) or ∀n ∈ N, n ≥ ⇒ P(n).
• Always label the steps that use the induction hypothesis.
• Please follow the same guidelines as Problem Set 2 for all proofs.
1. [4 marks] Special numbers. For each n ∈ N, define Fn = 22n + 1.
n−1
Prove that for all natural numbers n, Fn − 2 = Y Fi.
i=0 Hints:
• Please review product notation, including empty products, on page 16 of the course notes.
• For all .
2. [8 marks] Sequences. We define the following sequence of numbers a0,a1,a2 … recursively as:
a0 = 1, and for all
(a) [1 mark] What are the values of a0, a1, a2, and a3?
(b) [3 marks] Find and prove a non-recursive formula for an that is valid for all natural numbers n. That is, the statement you will prove should be of the form
∀n ∈ N, an =
By “non-recursive” we mean that the formula you use to fill in the blank should not involve any ai terms.
(c) [1 mark] Let’s now generalize the previous part. For every natural number k greater than 1, we define an infinite sequence ak,0,ak,1,… recursively as follows:
ak,0 = k, and for all
What are the values of a2,0, a2,1, a2,2, and a2,3? What are the values of a3,0, a3,1, a3,2, and a3,3?
(d) [3 marks] Find and prove a non-recursive formula for ak,n that is valid for all natural numbers k greater than 1, and all natural numbers n. Hint: as we saw in class, it’s easiest to handle multiple universal quantifications in a proof by induction by first letting one variable be arbitrary, and then doing induction on the other variable.
3. [11 marks] Properties of Asymptotic Notation.
Let f : N → R≥0. We define the cumulative sum of f, denoted Sumf, to be the function Sumf : N →
R≥0 defined as follows:
n
Sumf(n) = Xf(i) = f(0) + f(1) + ··· + f(n)
i=0
For example, we have previously proved in this course that if f(n) = n, then .
(a) [4 marks] Prove that for all f : N → R≥0, if f ∈ O(n), then Sumf ∈ O(n2).
b c b X X X
f(i) = f(i) + f(i) for all a ≤ c ≤ b. i=a i=a i=c+1
(b) [3 marks] Prove by induction that for all natural numbers .
(c) [4 marks] Using part (b), disprove the following claim: for all f,g : N → R≥0, if f(n) ∈ O(g(n)), then Sumf(n) ∈ O(n · g(n)).