Download Cardinality of infinite sets

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
• An infinite set, A, is countable iff
there is a one to one
correspondence between A and the
positive integers
– We refer to this cardinality, 𝐴 = ℵ0
– Last symbol is the Hebrew Aleph, read
Aleph null or aleph naught
• If this is not the case then the infinite
set is uncountable
– There is an hierarchy of alephs
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
• The function f(x) = 2x where x is a
positive integer
• This a mapping from positive
integers to positive even integers
• This mapping is one to one
• The definition of countable is now
met for the even positives, so the
cardinality of positive evens is ℵ0
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!
• What we now see is three infinite
sets with the same cardinality
• A is the positive evens
A  Z+  Q yet |A|  | Z+ |  |Q|
– All are ℵ0
• Thus, it is hard to think about
cardinality of infinite sets as exactly
the same as set size
• Much more similar to big O notation
• Where many details are largely ignored
Copyright © 2014 Curt Hill
Uncountable Sets
• If there are countable sets then
there must be uncountable sets
• The real numbers is such an infinite
set
– Cardinality ℵ1
• The book supplies a proof by
contradiction
– This will be similar
– See next slides
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 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 r1, r2, r3, …
• This sequence must have the
property that ri < ri+1
• Each rn has a decimal expansion
that looks like this:
0.d1d2d3d4d5…
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 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 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
• If A and B are countable then AB is
also countable
• If |A||B|  |A||B| then |A|=|B|
– Schröder-Bernstein Theorem
• The existence of uncomputable
functions
– Functions that cannot be generated by
program
• Continuum hypothesis
– No cardinality numbers between
ℵ0 𝑎𝑛𝑑 ℵ1
Copyright © 2014 Curt Hill
Exercises
• 2.5
– 1, 3, 5, 17, 23
Copyright © 2014 Curt Hill