Lecture 25 21 127 Concepts of Math 10 31 2012 Administrivia Exams were returned yesterday Solutions are posted on AnnotateMyPDF Grade cutoffs are below Note these are kinda loose There are clear cutoffs between a B and a C as you ll see from the distribution graph but there was no clear distinction between an A and a B nor a C and a D nor really even a D and an R I ll keep that in mind when I compute final grades For now though here is a rough idea of where you stand Remember that we also care about where in this distribution you fall I won t treat every C as the same in the final scheme necessarily Suppose your score is x then you earned A 177 x 210 B 145 x 176 C 100 x 144 D 79 x 99 R x 78 Notice that I am being very generous This is a response to the odd distribution of scores and also my overall perception of how well you have learned the material Countably Infinite Sets Infinity is weird I know Today we will focus on a certain size of infinity countably infinite sets By definition N is countably infinite we can enumerate its elements in order i e in some order that covers all possible elements by saying 1 2 3 Notice that we aren t claiming we can count all of the elements in any finite amount of time but we can identify a counting procedure that will eventually name every element Think of your favorite natural number even if it s a really large one we will say it eventually with our procedure We proved last week that Z is countably infinite by finding a bijection between Z and N We even proved it was a bijection by finding its inverse Think of the counting procedure for Z that is dictated by the bijection 0 1 1 2 2 3 3 No matter what integer you have in mind we will eventually get to it with this procedure An Explicit Bijection With the Hilbert Hotel example on Monday we proved that N N is countably infinite To do this we found an embedding of countably many lines of countably many people into a set of countably many rooms Essentially this is really only an injection from N N to N Realistically though that is overkill if anything we expected N N to be bigger than N so finding an injection showed that at worst N N is just as big as N Thus N N N More about this bigger and smaller stuff later Now let s discuss an explicit bijection between N N and N Define f N N N by x y N N f x y 2x 1 2y 1 Let s prove f is a bijection 1 Proof Injectivity follows by supposing f x y f u v and equating powers of 2 and odd factors Since a power of two is even and not odd unless that power of two is 20 1 it must be that x u Thu y v Surjectivity follows from a minimal criminal argument Suppose we have n that has no such representation If n is odd then duh n 20 n 1 n is such a representation If n is even then consider n2 Were we to have a representation of n2 then we would have one for n by assumption then n2 is a smaller counterexample to the claim This is a contradiction so there is no least counterexample This formally proves that N N N Furthermore this is an efficient packing of the countably many countable conventions into the Hilbert Hotel without leaving any empty rooms A Discussion Of Axioms Definitions and Results Here s the thing we glossed over many details about what constitutes a definition as opposed to a theorem a result that needs proven from fundamental assumptions By definition at least in our context an injection and a surjection from A to B in that direction mind you constitute sufficient proof of equal cardinalities which guarantees a bijection Likewise an injection from A to B and one from B to A is sufficient to guarantee a bijection between A and B It is not totally obvious though why this should be true Say we have an injection from A to B and one from B to A Does this guarantee a bijection between the two sets One would hope But this isn t a proof This result is known as the Cantor Schroeder Bernstein Theorem Yes that is a theorem it is not trivial One of the proofs is in fact constructive that is it provides an algorithmic method for constructing a bijection between A and B assuming the existence of two injections f A B and g B A For our purposes there is no need to separate this out as a theorem let alone one with a constructive proof It is sufficient to consider injections and surjections and their consequences vis a vis cardinalities as definitions these results feel intuitive and we can accept them Just realize though that we are basing them on rigorous mathematical knowledge In essence the real issue is that we pre supposed any two sets A and B can have their cardinalities compared in some meaningful way mathematically speaking That is for any A and B we can somehow declare that A B or B A makes sense or both perhaps if the sets are of equal size But how does that really make sense Can we guarantee one such comparison or maybe both will always apply for any two given sets It s not a trivial consideration In any event this is why we feel it s far more interesting and ultimately mathematically rigorous to provide an explicit bijection between N N and N Sure we already have an injection the powers of primes function we came up with in lecture on Monday when we faced countably many conventions of countably many people seeking vacant rooms and we could very easily come up with an injection the other way just map n to 1 n say so then C S B guarantees a bijection The proof would even construct the bijection However an explicit definition of a function and a formal proof that is hardly more than algebraic in nature with a dash of induction sprinkled on top feels far more convincing Another Proof Consider N N as an infinite lattice A proof of countability amounts to a description of a path through this lattice that hits every ordered pair of naturals once and only once The dovetailing method accomplishes this we just walk along the finite diagonals one by one It is not totally trivial to write down an explicit formula that describes this bijection but it is possible if you just play around with it for a little while assuming even that though it is not trivial to prove this function is a bijection In the end though this is a more helpful and memorable counting method that lists out the elements of N N …