Homework on Induction Proofs with Recursive Formulas in Discrete Mathematics, Assignments of Discrete Structures and Graph Theory

Download Homework on Induction Proofs with Recursive Formulas in Discrete Mathematics and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity! CMSC 250 - Summer 2008 - Homework 6 Due Tuesday, July 1 Issued June 26, 2008 For the *Guided Question with Prof. D. Math* I encourage (but do not mandate) you to let Prof D. Math guide you, and to limit collaboration to understanding the examples and solutions in the referred lecture notes and textbook pages, and only seeing the Instructor or the TA if you are further lost. 1. (20 points) *Guided Question with Prof. D. Math* The Problem Good to see you again! Your mission, should you choose to accept it, is to: Consider the recursive formula for the sequence a0 = 1,∀n ≥ 1 [ an = 1 + 2 ∗ n−1∑ i=0 ai ]. Prove that ∀n ≥ 0 [ an = 3n ]. You must give two proofs: One that uses Simple Induction and one that requires Strong Induction. I have talked to Peter and all you must do is provide the two solutions to the problem to receive full credit for the problem. You do not have to answer any of the questions. However, both Peter and I recommend that you follow the guide and we will give you feedback (without point deductions) on the questions if you choose to submit answers to the questions with the problem. Ready? Let’s begin! The Guide Here we will give two proofs to illustrate the differences between Simple Induction and strong Induction. Here to illustrate the subtle points of induction we will examine. Note: It might help to first read the entire guide and then work through it. When thinking about the number of needed base cases, it is often learned when we do the Inductive Step. Hence, it is a good idea to leave some space for additional base cases at your Base Case part of the proof if you later find you need additional base cases. • For both proofs you will need at least 1 base case. (This is to satisfy the ‘‘Base Case’’ part of the Principle of Mathematical Inducution.) Write the value of n for that base case and the proof for that one base case. 1 • Now it is a good idea to get the initial Inductive Hypotheses that you will need for each proof. Here write the Ind. Hyp for both proofs, and label which is the Hypothesis for the Simple Inductive proof and which is for the Strong Inductive proof. For now, your Base case and your Inductive Hypotheses may be incomplete. We will fill in the holes shortly! Simple Induction Proof Now we will look at the Simple Inductive proof. Here is the clever part of the inductive step: Inductive Step: Show for n = k ak = 1 + 2 ∗ k−1∑ i=0 ai (1) = 1 + k−1∑ i=0 2ai (2) = 1 + k−2∑ i=0 (2ai) + 2ak−1 (3) = 1 + 2 ∗ k−2∑ i=0 (ai) + 2ak−1 (4) = ak−1 + 2ak−1 (5) = 3ak−1 (6) Look at this closely. Note: In your proof you do not have to show all of the constant manipulations. The critical steps are (1), (3), (5) and (6), which are the ones you should at least show. • Look at step (1). To get step (1) I used the recursive formula I was given. What must be true for this step to work? • Look at step (5). What did I do to get from (4) to (5)? What must be true for this step to work? • Based on your conclusions above, what is the minimal number of base cases that you need for this proof? Why? • There is a property/assumption that is needed in your Inductive Hypothesis. What is it? Why? (Hint: This property follows from the ‘‘Base case’’ part of your proof.) Great! All that is left to this Inductive Step is to use the Simple Induction Hypothesis. • Now finish the Inductive Step. Note that you only need the Simple Inductive Hypothesis. 2