CS 1050B: Constructing Proofs
Supplementary Exercises 3 : Comprehensive
Nov 29, 2006
Rosen Chapter 1 : Logic and Proofs
1. Chapter 1 Supplementary P.40
Assuming the truth of the theorem that states that √nis irrational whenever nis a positive
integer that is not a perfect square, prove that √2 + √3 is irrational.
2. 1.7 P.34
Prove that between every two rational numbers there is an irrational number.
Chapter 4 : Induction and Recursion
Chapter 7 : Recurrence Relations
3. Chapter 4 Supplementary P.4
Use mathematical induction to show that
1
1·3+1
3·5+· ·· +1
(2n−1)(2n+ 1) =n
2n+ 1
whenever nis a positive integer.
4. 4.2 P.12
Use strong induction to show that every positive integer ncan be written as a sum of distinct
power of two, that is, as a sum of a subset of the integers 20= 1,21= 2,22= 4, and so on.
[Hint :For the inductive step, separately consider the case where k+ 1 is even and where it is
odd. When it is even, note that (k+ 1)/2 is an integer.]
5. Give a recursive definition of the sequence an,n= 1,2,3,··· if
a) an= 5
b) an= 10n
c) an=n+ (−1)n
d) an=n!(n2+ 2)
(This was the first problem from quiz 1. Check out the solution if you are stuck. We
are expecting that you will be able to write up the solution in the same way during the
final exam.)
6. Find the solution to each of these recurrence relations and initial conditions. Use an iterative
approach
1
