Lecture 5 21 127 Concepts of Math 09 07 2012 Administrivia Homework 1 will be returned on Tuesday in recitation Homework 2 has been posted and is due next Thursday in recitation Induction Wrap up Let s summarize some of our work on inductive arguments explain what we ve accomplished so far and think about where we re going Summary We should all be comfortable with the idea of an inductive argument When we want to prove some fact that depends on an integer usually just a natural number even as well as some previous instance of that fact we can explain this via an inductive argument We can explain the first case or several cases if need be and then use a general variable to express the relationship of some instance of the fact to previous cases Between these two arguments as long as they are put together appropriately we can show that the fact holds true for all of the instances that makes sense We saw several examples of how to work through an inductive argument to develop a guess at a fact and then go back and prove that fact somewhat rigorously For example we used some geometric intuition to come up with the conjecture n X 2k 1 n2 k 1 We noticed that a square grid is compose of several L shaped corners of increasing size We then went back and gave an algebraic proof of this fact using an inductive argument With some situations though we noticed that an instance of the fact depended on several previous instances For example when we were counting the number of distinct domino tilings of a 2 n grid we came up with the relationship T n T n 1 T n 2 Here each fact depended on two previous instances so we would require two initial instances or base cases to get our inductive argument off the ground Somewhat similarly when you considered the Takeaway game in recitation you needed to refer to a previous instance of the game but you didn t know exactly which version you would need Questions What s really going on there It s hard to say We gave you a heuristic description of mathematical induction an intuitive explanation of why it works Think about the domino analogy which is explained more thoroughly in Chapter 2 of the textbook If we can get the first domino to fall and we know that every domino falls into the next one then surely the whole line will fall right This makes sense but you should be a little wary when infinity come into play Furthermore how do we explain the domino tiling problem Or the Takeaway game What is the inductive argument there Why does it make sense Despite what you may or may not think these are difficult questions mathematically speaking We want to answer them thorougly We want to be as rigorous as possible We want you to understand where 1 we re coming from though which is why we have still been working on these types of arguments and why there are several such problems on your homework for next week We are helping you out and guiding you through the proofs as much as possible Soon enough we will understand how and why all of this works and then you will be pretty much on your own Motivating Example To give you an indication of why we need to set these ideas on firm mathematical foundations let s consider the following proof What is wrong with the following spoof that all pens have the same color Spoof We claim that all pens have the same color We will prove this by showing that a set of pens of any size has only one color represented amongst those pens We will provide an inductive argument for this claim by inducting on the size Consider a group of pens with size 1 Since there is only 1 pen it certainly has the same color as itself Now suppose that any set of n pens has only one color represented inside the group Take any set of n 1 pens Line them up on a table and number them from 1 to n 1 left to right Look at the first n of them i e look at pens 1 2 3 n This is a set of n pens so by assumption there is only one color represented in the group We don t know what color that is yet Then look at the last n of the pens i e look at pens 2 3 n 1 This is also a set of n pens so by assumption there is only one color represented in this group too Now pen 2 happens to belong to both of these sets Thus whatever color pen 2 is that is also the color of every pen in both groups Thus all n 1 pens have the same color By induction this shows that any group of pens of any size has only one color represented Looking at the finite collection of pens in the world then we should only find one color What went wrong here Sets We ve been dancing around the idea of what a proof is lately and it s time to finally settle in and develop some fundamental concepts and notation In a way we are starting from the beginning from axioms We will develop all of the necessary tools and results to not only explain what a good proof is but also to show why certain proof strategies are valid We won t just accept that a technique is okay we want to know how it works This will be decidedly different from what we have done so far We will need to do some abstract thinking We will need to incorporate new definitions and symbols into our vocabulary We will need to practice expressing our thoughts in this new mathematical language The first new fundamental mathematical object we will talk about is a set Definition and Examples Definition A set is a collection of objects that have a common well defined property The objects contained in a set are called elements of the set The mathematical symbol represents the phrase is an element of and represents is not an element of 2 We already have a few standard sets at our disposal N 1 2 3 Z 2 1 0 1 2 na o Q where a b Z and b 6 0 b R real numbers Note The order of the elements does not matter Also repeating an element has no effect These are all valid ways of defining the same set V a e i o u V vowels in the English language V u e i a o V u e i u o a u e i a We could also define a set like B Major League Baseball teams from the 2012 season and this is much better than having to write out the 30 elements of the set What about this set C Major League Baseball teams from the 2011 season It …