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
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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, – 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 cardinalityCopyright © 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
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 3 4 51 1/1 2/1 3/1 4/1 5/12 1/2 2/2 3/2 4/2 5/23 1/3 2/3 3/3 4/3 5/34 1/4 2/4 3/4 4/4 5/45 1/5 2/5 3/5 4/5 5/5Start 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!• 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
• 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
• 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 non-nine 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 AB
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
Copyright © 2014 Curt Hill
Exercises• 2.5
– 1, 3, 5, 17, 23
Copyright © 2014 Curt Hill
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