Lecture 14 21 127 Concepts of Math 10 01 2012 Administrivia Exams will be returned tomorrow in recitation It is logistically infeasible to return them in lecture I don t know anything about averages or grades or any other stats Regrade Policy After you receive your exam take a couple of days to look them over Solutions are already posted so you can compare and see what you think If there are minor issues the points don t add up you can t read a number or comment etc then feel free to take that issue to your TA and have them sort it out If you just don t understand why a problem was graded a certain way feel free to ask your TA or me about that I have copies of the rubric and can explain what we felt was important in the problem and whether you addressed that in your solution If you actually have a grading complaint I think my answer deserves more points I think you didn t understand what I meant in this part you misread my solution etc you must do the following For each problem you want to be considered for a re grade write something about why you think your solution deserves more points Explain how you think you understood the question and what knowledge you demonstrated in the solution Explain why you think that demonstration is worthy of some grade This might be anywhere from a couple of sentences if you think we just misread something or made a simple error to a paragraph or two if you need to re explain your solution I will not accept something like You were too harsh I probably won t even re read your solution if you aren t willing to put in the time to think about the problem and what you wrote You can do this anytime this week hand them in to me or your TA who will bring them to me but next Monday s lecture will be the absolute latest I will accept them I will personally read them and may or may not change things Then you can pick them up from your TAs Meanwhile Homework 4 is due on Thursday It s shorter than the others for your benefit Come to office hours for help 1 Induction Induction should apply when we are attempting to prove some property holds for all of the natural numbers That is we will use induction when we want to prove something like n N P n where P n is some proposition that depends on n Proposition The Principle of Specific Mathematical Induction Let P n be a logical statement Assuming 1 P 1 holds and 2 n N P n P n 1 then the statement n N P n is true Proving the Principle of Induction Note This is interesting but not essential to your success in the course Before we do this let s recall some definitions These were seen in Lecture 9 Definition A set I is called inductive provided 1 1 I 2 If n I then S n I as well In the context of N the successor is S n n 1 This has a set theoretic definition for other contexts and it matches up with the definition of natural numbers as sets themselves that we briefly mentioned in Lecture 9 too Intuitively we know that N is inductive as is N 0 Z etc Definition The set of all natural numbers is the set of elements that belong to every inductive set N I I S S is inductive Put another way N is the smallest inductive set Note Why is the intersection above non empty That is how do we know there are actually any inductive sets to begin with Well we have to assume this We have to make an axiomatic assumption somewhere in our theory Now let s prove the statement of the Principle of Mathematical Induction PMI Proof Let P n be a logical statement Define the set T by T x N P x That is T is the set of natural numbers for which the logical statement is true By assumption 1 T We assume P 1 2 Let n T be arbitrary By definition P n holds Then by assumption P n 1 follows We assume P n P n 1 for any n This means n 1 T This means we have proven the implication n T n 1 T Thus T is an inductive set Recall the definition of Inductive Set from Lecture 9 By the definition of N then N T Recall that we defined N as the intersection of all inductive sets so N I for every inductive set I Also by the definition of T using set builder notation T N Therefore T N The main idea here is this When we prove something by induction we are proving that the set of instances for which the claim holds is equal to the set N In the proof of PMI that set T is the set of instances for which the claim holds 3 Using Induction to prove something This is the important part Do not worry too much about the proof of induction I am far more interested in your ability to use induction to prove things Specifically I want to see you follow the format for an induction proof that I have outlined below and you should understand why this template works In practice when we prove something P n by induction we are actually showing that the hypotheses of the PMI hold This allows us to necessarily deduce its conclusion which is to say that the set of instances for which the claim holds is equal to the set N That is given P n we are showing that 1 P 1 holds and 2 k N P k P k 1 holds and concluding that n N P n holds Template for a Proof by Induction Goal Prove that n N P n Proof Let P n be the proposition We will prove n N P n by induction on n Base Case Observe that P 1 holds because Induction Hypothesis Let k N be arbitrary and fixed Suppose P k holds Induction Step Deduce that P k 1 also holds By PMI it follows that n N P n 4 Comments and common pitfalls Sometimes the claim is defined for you in the problem statement However it is not always defined explicitly as P n In that case referring to a proposition P n later on has no meaning You need to say something like Let P n be the claim defined above Explicitly state that you are using mathematical induction and state which variable to which you are applying induction Do not expect a reader to just understand you are going to use induction a priori There might be multiple variables floating around Tell the reader which one is inducted Be as explicit and thorough as possible in the Base Case Do not just write out what P 1 means and expect the …