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Maggie Johnson
Handout #8
CS103B
Infinite Sets and Countability
Key topics:
* Infinite Sets
* Diagonalization
* Russell's Paradox
* The Beginnings of Theoretical Computer Science
Infinite Sets
In a
finite
set, we can always designate one element as the first member, another as the second
member, etc.
If there are k elements in the set, then these can be listed in the order we have
selected:
s
1
,
s
2
,
...,
s
k
Therefore, a finite set is one which can be enumerated by the positive integers from 1 up to some
integer k.
More precisely, A is finite if there is a positive integer k such that there is a onetoone
correspondence between A and the set of all natural numbers less than k.
We define onetoone
correspondence between the elements of a set P and the elements of a set Q if it is possible to pair
off the elements of of P and Q such that every element of P is paired off with a distinct element of Q.
We are defining a function that maps the elements of P to the elements of Q; this function must be
both onetoone and onto (or bijective for you function enthusiasts out there).
If a set is infinite, we may still be able to select a first element s
1
, and a second element s
2
, but we
have no limit k.
So the list of chosen elements may look like this:
s
1
,
s
2
,
s
3
......
Such an infinite set is called
denumerable
.
Both finite and denumerable sets are
countable
sets
because we can count, or enumerate the elements in the set.
Being countable, however, does not
always mean that we can give a value for the total number of elements in the set; it just means we
can say "Here is a first one, here is a second one.
....".
In other words, a denumerable set is one
where we can define a onetoone correspondence between the elements in the set and
Z
+
.
Thus, the set
Z
+
is, in a sense, the most basic of all infinite sets.
The reason for this is that the one
toone correspondence can be used to "count" the elements of an infinite set.
If f is a onetoone and
onto function from
Z
+
to some infinite set A, then f(1) can be designated as the first element of A,
f(2), the second, and so forth.
Because f is onetoone, no element is ever counted twice; and
because f is onto, every element of A is counted eventually.
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View Full DocumentTo prove denumerability, we need only exhibit a counting scheme, i.e., if starting from a particular
element, we can sequentially list all the elements in the list, (because such a listing will yield a one
toone correspondence between the elements in the set and
Z
+
).
The counting scheme is the
function that maps
Z
+
to some other infinite set.
Example 1
The set of positive even integers {0, 2, 4, 6, 8.
..} is denumerable because there is an obvious oneto
one correspondence between these integers and the positive integers (2k).
The set of all integers,
positive and negative is denumerable because we can list them as follows: {0, 1, 1, 2, 2, 3, 3 .
..}
(f(n) = n/2 if n is even, (n1/2) if n is odd).
But these examples seem to go against common sense.
The set of positive even integers must be 1/2 the size of the set of positive integers; the set of all
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 Winter '08
 SAHAMI,M
 Computer Science

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