PDF-Some Examples of Proof by Induction 1 By induction prove that

0n for 0n For 0nlet Pn 1470nBasis step 0Pis true since002Inductive step For0n since if0n then 22 2 By induction for 1nprove that if the plane cut by n distinct lines the interior of the regions bound. EECS . 203. : Discrete Mathematics. Lecture . 11 . Spring 2015. 1. Climbing the Ladder. We want to show that ∀. n. ≥1 . P. (. n. ) is true.. Think of the positive integers as a ladder.. 1, 2, 3, 4, 5, 6, . . .. (chapter 4.2-4.4 of the book and chapter 3.3-3.6 of the notes). This Lecture. Last time we have discussed different proof techniques.. This time we will focus on probably the most important one. – mathematical induction.. Cappello. Mathematical Induction. Goals. . Explain & illustrate construction of . proofs of a variety of theorems using mathematical induction.. Copyright © Peter . Cappello. Motivation. Mathematics uses 2 kinds of arguments:. (chapter 4.2-4.4 of the book and chapter 3.3-3.6 of the notes). This Lecture. Last time we have discussed different proof techniques.. This time we will focus on probably the most important one. – mathematical induction.. Can you find an ordering of all the n-bit strings in such a way that . two consecutive n-bit strings differed by only one bit?. This is called the Gray code and has many applications.. How to construct them?. and Other Forms of . Induction Proof. Sanghoon Lee & Theo Smith. Honors 391A: Mathematical Gems. Prof. . Jenia. . Tevelev. March 11, 2015. How does induction work?. 1.) Base Case: Show the First Step Exists. . Predicate. Logic. LN . chapters. 3,4. STV 2016/17. Testing vs verification. Testing .  practical, but incomplete.. Verification: here it means . proving. that the program is correct, hence complete.. 473/474. How (not) to do an induction proof. A . B (A implies B) means that whenever A is true, B is true also. The only way . A .  . B can be false is when A is . true. and B is . false. .. The inverse, B . -Remember: Ask lots of questions on Piazza, ask others for help, Google whatever you need to. -Only requirement: write your solutions by yourself (without extensive notes). -Today: Recursion refresher. Introduction. Proof by mathematical induction is an extremely powerful tool for proving mathematical statements. As we know, proof is essential in . Maths. as although something may seem to work for a number of cases, we need to be sure it will work in every case. Mathematics. 1. Mathematical . vs. Strong Induction . To prove that . P. (. n. ) is true for all positive . n. .. Mathematical. induction:. Strong. induction:. 2. Climbing the Ladder (Strongly). We want to show that ∀. Strong Induction EECS 203: Discrete Mathematics 1 Mathematical vs Strong Induction To prove that P ( n ) is true for all positive n . Mathematical induction: Strong induction: 2 Climbing the Ladder (Strongly) This Lecture. Last time we have discussed different proof techniques.. This time we will focus on probably the most important one. – mathematical induction.. This lecture’s plan is to go through the following:. Why is it a legitimate proof method?. How to use it?. Z all integers (whole numbers). Z. +. the positive integers. Z. -. the negative integers. N Natural . numbers: non-negative integers.