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Please help me with question 2 to question 5. Thank you.

Please help me with question 2 to question 5. Thank you.

CSC236 Fall 2016


Assignment #1: induction


due October 7th, 10 p.m. The aim of this assignment is to give you some practice with various forms of induction. For each question


below you will present a proof by induction. For full marks you will need to make it clear to the reader that


the base case(s) is/are veri ed, that the inductive step follows for each element of the domain (typically the


natural numbers), where the inductive hypothesis is used and that it is used in a valid case.


Your assignment must be typed to produce a PDF document (hand-written submissions are not acceptable). You may work on the assignment in groups of 1, 2, or 3, and submit a single assignment for the entire


group on MarkUs


1. Consider the Fibonacci-esque function g : 8






g (n) = 3;




:g(n 2) + g (n if n = 0


if n = 1


1) if n > 1 Use complete induction to prove that if n is a natural number greater than 1, then 2n=2  g (n)  2n .


You may not derive or use a closed-form for g (n) in your proof.


2. Suppose B is a set of binary strings of length n, where n is positive (greater than 0), and no two


strings in B di er in fewer than 2 positions. Use simple induction to prove that B has no more than


2n 1 elements.


3. De ne T as the smallest set of strings such that:


(a) "b" 2 T


(b) If t1 ; t2 2 T , then t1 + "ene" + t2 2 T , where the + operator is string concatenation.


Use structural induction to prove that if t 2 T has n "b" characters, then t has 2n 2 "e" characters. p 4. On page 79 of the Course Notes the quantity  = (1 + 5)=2 is shown to be closely related to the


Fibonacci function. You may assume that 1:61803 <  < 1:61804. Complete the steps below to show


that  is irrational.


(a) Show that ( 1) = 1.


(b) Rewrite the equation in the previous step so that you have  on the left-hand side, and on the


right-hand side a fraction whose numerator and denominator are expressions that may only have


integers, + or , and . There are two di erent fractions, corresponding to the two di erent factors in the original equation's left-hand side. Keep both fractions around for future consideration. 1 (c) Assume, for a moment, that there are natural numbers m and n such that  = n=m. Re-write the


right-hand side of both equations in the previous step so that you end up with fractions whose


numerators and denominators are expressions that may only have integers, + or , m and n.


(d) Use your assumption from the previous part to construct a non-empty subset of the natural


numbers that contains m. Use the Principle of Well-Ordering, plus one of the two expressions for


 from the previous step to derive a contradiction.


(e) Combine your assumption and contradiction from the previous step into a proof that  cannot


be the ratio of two natural numbers. Extend this to a proof that  is irrational.


5. Consider the function f , where 3  2 = 1 (integer division, like 3==2 in Python): ( f (n) = 1


if n = 0




f (n  3) + 3f (n  3) if n > 0 Use complete induction to prove that for every natural number n greater than 2, f (n) is a multiple of


7. NB: Think carefully about which natural numbers you are justi ed in using the inductive hypothesis


for. 2


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