Cardinality of Infinite Sets There be monsters here
Cardinality of Infinite Sets There be monsters here! At least serious weirdness! Copyright © 2014 Curt Hill
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
1 1 2 3 4 5 2 Count ‘em 3 4 5 1/1 2/1 3/1 4/1 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/3 1/4 2/4 3/4 4/4 5/4 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 … Copyright © 2014 Curt Hill
Very Interesting! • Copyright © 2014 Curt Hill
Uncountable Sets • Copyright © 2014 Curt Hill
Reals Uncountable 0 • There exists a theorem that states 0. 9999… is the same as 1. 0 – The idea of the proof is that as the 9 s 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 ri < ri+1 • Each rn 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, rn and rn+1 • These two must be different at some di – If they are not we have numbered identicals – We also disallow that the lower one is followed by infinite 9 s and the higher one by infinite zeros which would be two representions of the same number Copyright © 2014 Curt Hill
Reals Uncountable 3 • Now rn must have a non-nine following di call it dj – Otherwise we violated the no identicals rule • Create a new real rk that is rn with dj incremented by 1 • We now have rn < rk < rn+1 which contradicts our original assertion • In fact we can insert an infinity of such numbers by incrementing the digits after dj Copyright © 2014 Curt Hill
Reals Uncountable Addendum • In the last screen there was the argument that there must be a nonnine in the sequence • The symmetrical argument is that there must be a non zero following the rn+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
- Slides: 18