Lecture 26 21 127 Concepts of Math 11 02 2012 Countably Infinite Results Theorem A countably infinite union of countably infinite sets is also countably infinite This is the most important result of cardinality It just is That result is the most important result of cardinality It just is Like sure the reals is uncountable that s important This is practical They will see that in an analysis course that s an important result Right Will Gunther This is a very surprising result and extremely helpful in other areas of math The fact that we can combine a whole bunch of infinite sets and get the same size of infinity afterwards is very helpful to know Many fundamental results in real analysis are based on this fact Proof Sketch We can biject each set to N This lets us number the elements of each set in order according to their correspondence with N Since we have countably infinitely many sets as well we can doubly number those elements with which set they came from This lets us represent each element of the big union by an ordered pair i j where ai j is the j th element of set Ai Now assuming the sets are pairwise disjoint this defines a bijection from the big union Ai to the set i N N N thus proving countability If the sets are not pairwise disjoint though our correspondence as ordered pairs just fails to be an injection because multiple ordered pairs correspond to the same element That is suppose the element appeared as the 1st element of A2 and the fourth element of A5 then the pairs 2 1 and 5 4 both correspond to It is a surjection though from N N to the big union so we have shown that the union is at most as big as N N This proves countability Example 1 The set of all powers of primes Remember the Hilbert Hotel when we accomodated infinitely many lines of people that were infinitely long We sent people to rooms corresponding to powers of primes For every n N define pn to be the n th prime number We already proved way back in the first couple weeks of class that there are infinitely many prime numbers so we can do this Then for every n N define An pkn k N which is the set of all powers of the n th prime The result above says that An all powers of primes n N is countably infinite as well Indeed we should have expected that because that union is just a subset of the natural numbers even Example 2 The set of all finite binary strings A binary string is just an ordered list of 0s and 1s A finite binary string is one that is of finite length For example the following are all finite binary strings and thus elements of the set we will define in a second 0 1 101010 10000000000000000001 For every n N let s define Fn to be the set of all binary strings of length n For instance F1 0 1 F2 00 01 10 11 F3 000 001 010 100 011 101 110 111 1 and so on Notice that Fn 2n Try to prove that Then define the set of all finite binary strings by F Fn n N An element of F must have come from some set in the big union this means that an arbirtrary element x F is some binary string with some finite length That length could be a huuuuuuuge number but it is finite This points out the distinction between allowing something to be arbitrarily large but finite and allowing something to be infinite The point of this example is that F is countably infinite according to the theorem Contrast this with the set S of all infinite binary strings which is uncountably infinite You considered this on the Exam 2 Prep Questions by finding a bijection between S and P N and we know P N to be uncountably infinite Passing Off To A Limit Remember the Theorems we proved last time If A and B are countably infinite then so are A B and A B We also said that we can prove by induction on the number of sets in the union product that Y A1 A2 An Ai and Ai A1 A2 An i n i n are both countably infinite as well for any n N What do these results tell us if anything about A1 A2 A3 Ak k N and A1 A2 A3 Y Ak k N That is what happens when we try to jump the limit from having a finite union product of arbitrarily large size but still finite to having an infinite union product Can we make necessary conclusions Can we find counterexamples The main idea is that passing to a limit does create some mathematical object but we can t necessarily pre suppose that this object has the exact same properties as all of the objects in the sequence that defines that object Think about the finite sets n for every n Each of them is finite but in the limit we get N which is not finite So yes we do get some object another set but it doesn t have to have the same properties The first Theorem we proved today says that passing to the limit in the union definitely preserves countability As we will see below the product definitely does not preserve countability In fact even an infinite product of finite sets is uncountable Yikes A similar notion appears in calculus I promised I would not use calculus this semester but there is such a natural relationship between these ideas and you all asked about it in class so I felt like I had to If you don t get anything out of this no worries if you do though try to remember this connection and think about how it might fundamentally change your view of everything you learned in calculus Consider a limit something like lim x 1 0 x 2 In what sense is this limit equal to 0 Why would we as mathematicians over the years choose to define limits in this way Formally this limit makes sense because of the quantified definition of a limit Let P be the set of positive real numbers Then the definition of limit applied to this example says P M N n N n M 1 x That is for any small positive threshold 0 we can find a specific cutoff point a large natural number M that depends on somehow such that for every point after M the function x1 falls within that threshold of the limit point zero 1 0 That s not what s going on Notice that this is very different than saying some nonsense like We never actually get to plug in the end of the limit and evaluate it The limit is defined in terms of quantifications some things that are happening for arbitrarily large values but not for an infinite value Uncountably Infinite Results Theorem A countably …