Copyright © 2014 Curt Hill Cardinality of Infinite Sets There be monsters here! At least serious weirdness!

Cardinality Recall that the cardinality of a set is merely the number of members in a set This makes perfect sense for finite sets, but what about infinite sets? We may compare the sizes of such infinite sets by attempting a one to one correspondence between the two It gets a little weird here and our intuition does not always help Copyright © 2014 Curt Hill

Some Definitions Sets A and B have the same cardinality iff there is a one to one correspondence between their members Finite sets are obviously countable The notation for cardinality is the same as absolute value If A = {1, 3, 4, 5, 9} then |A|=5 Copyright © 2014 Curt Hill

Infinite Countable Sets Copyright © 2014 Curt Hill

Counter Intuitive We would normally think that: –If A  B then |A|< |B| This is true for finite sets but not necessarily for infinite sets Consider the positive even integers It is a subset of the positive integers Yet it is one to one with the positive integers –Thus is a countably infinite set and has the similar cardinality Copyright © 2014 Curt Hill

Countable Copyright © 2014 Curt Hill

Hilbert’s Grand Hotel This paradox is attributed to David Hilbert There is a hotel with infinite rooms Even when the hotel is “full” we can always add one more guest –They take room 1 and everyone else moves down one room This boils down to the notion that adding one to infinity does not change infinity:  +1 =  –Recall that infinity is not a real number Copyright © 2014 Curt Hill

Another Countable The rationals are countable as well Use a matrix of integers –One axis is the numerator –The other the denominator –Duplicate values ignored In a diagonal way enumerate each rational –That is set them in one to one with positive integers Copyright © 2014 Curt Hill

Count ‘em Copyright © 2014 Curt Hill /1 2/1 3/1 4/1 5/1 2 1/2 2/2 3/2 4/2 5/2 3 1/3 2/3 3/3 4/3 5/3 4 1/4 2/4 3/4 4/4 5/4 5 1/5 2/5 3/5 4/5 5/5 Start at 1/1 and diagonally count each non-duplicate. 1/1 is 1, 2/1 is 2, 1/2 is 3, 1/3 is 4, 3/1 is 5 …

Very Interesting! Copyright © 2014 Curt Hill

Uncountable Sets Copyright © 2014 Curt Hill

Reals Uncountable 0 There exists a theorem that states … is the same as 1.0 –The idea of the proof is that as the 9s go to infinity the limit of the difference is zero –In other words however small you want the difference between two distinct reals to be we can make the difference between these two less An uncountable set cannot be a subset of a countable set Copyright © 2014 Curt Hill

Reals Uncountable 1 Assume that the reals between 0 and 1 are countable Then there is a sequence r 1, r 2, r 3, … This sequence must have the property that r i < r i+1 Each r n has a decimal expansion that looks like this: 0.d 1 d 2 d 3 d 4 d 5 … where each d is a digit Copyright © 2014 Curt Hill

Reals Uncountable 2 Next look at any adjacent pair of reals, r n and r n+1 These two must be different at some d i –If they are not we have numbered identicals –We also disallow that the lower one is followed by infinite 9s and the higher one by infinite zeros which would be two representions of the same number Copyright © 2014 Curt Hill

Reals Uncountable 3 Now r n must have a non-nine following d i call it d j –Otherwise we violated the no identicals rule Create a new real r k that is r n with d j incremented by 1 We now have r n < r k < r n+1 which contradicts our original assertion In fact we can insert an infinity of such numbers by incrementing the digits after d j Copyright © 2014 Curt Hill

Reals Uncountable Addendum In the last screen there was the argument that there must be a non- nine in the sequence The symmetrical argument is that there must be a non zero following the r n+1 Since we disallowed a …9999… followed by …0000… we can shift the argument to the second rather than first Copyright © 2014 Curt Hill

Results Copyright © 2014 Curt Hill

Exercises 2.5 –1, 3, 5, 17, 23 Copyright © 2014 Curt Hill