Problem_Set_04_Plus

Problem_Set_04_Plus - ArsDigita University Month 2:...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Problem Set 4 – Induction and Recurrence Equations 1. What’s wrong with the following proofs by induction? a. Every binary string contains identical symbols. The proof is by induction on the size of the string. For n= 0 all binary strings are empty and therefore identical. Let X = b n b n-1 …b 1 b 0 be an arbitrary binary string of length n+ 1. Let Y = b n b n-1 …b 1 and Z = b n-1 …b 1 b 0 . Since both Y and Z are strings of length less than n+ 1, by induction each contains identical symbols. Since the two strings overlap, X must also contain identical symbols. b. Any amount of change greater than or equal to twenty can be gotten with a combination of five cent and seven cent coins. The proof is by induction on the amount of change. For twenty cents use four five-cent coins. Let n > 20 be the amount of change. Assume that n = 7 x + 5 y for some non-negative integers x and y. For any n > 20, either x > 1, or y > 3. If x > 1, then since 3(5) – 2(7) = 1, n+ 1 = 5( y +3)+7( x –2). If y > 3, then since 3(7) – 4(5) = 1, n+ 1 = 7( x +3)+5( y –4). In either case, we showed that n+ 1 = 7 u + 5 v where u and v are non-negative integers. 2. Prove by induction that: a. The n th Fibonacci number equals (1/ 5)[(1/2 + 5/2) n – (1/2 – 5/2) n ], where F 0 = 0 and F 1 = 1. b. The sum of the geometric series 1 + a + a 2 + … + a n equals (1– a n +1 )/(1– a ), where a does not equal one. c. 21 divides 4 n+ 1 + 5 2n- 1 d. The determinant of the n by n square matrix below is equal to the determinant of B, where B is a matrix of m by m ( m<n ), and I is the m n identity matrix.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

Problem_Set_04_Plus - ArsDigita University Month 2:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online