Download Largest Contiguous Sum

INFIINITE SETS
CSC 172
SPRING 2002
LECTURE 23
Cardinality and Counting
The cardinality of a set is the number of elements in
the set
Two sets are equipotent if and oly if they have the
same cardinality
The existence of a one-to-one correspondence
between two sets proves that they are equipotent
Counting is just creating a one-to-one
correspondence between set and the set of integers
from 1 to some number n
Example

1

2

3

4

5
Finite & Infinite Sets
Can you create a one-to-one correspondence between
a set and a proper subset of itself?
If you can, you have a solution to x = x + y, where x
is the cardinality of the set and y >= 1 is the
cardinality of the stuff you removed
Impossible with finite sets
Possible with infinite sets
Technically, an infinite set is a set where there exists
a one-to-one correspondence between the set itself
and a proper subset
Example
Let N be the set of integers > 0 {1,2,3,…}
Clearly N –{1} is a proper subset of N {2,3,4,….}
We can create a 1:1 correspondence between the two
sets by matching each element x  N with element
x+1  N - {1}
Therefore, N is an infinite set.
Countable Infinity
Once we have an infinite set, we can prove another
set infinite by creating a one-to-one
correspondence between the known-infinite set and
a subset (possibly the whole set) of theother set
For example, The set of all integers Z {.., -2,1,0,1,2,…} contains N, (N is 1:1 with itself) so Z is
infinite.
Z and N
Z and N are actually equipotent.
What is the 1:1?
{1,2,3,4,…}
{…-3,-2,-1,0,1,2,3,…}
Each element x  N maps
To (x/2) if x is odd
Or –(x/2) if x is even
…
4,
3,
2,
{1
{…,-2,-1,0,1,2,..}
The mapping goes both ways, so Z is equipotent with N
Pairs to N
…
(1,5) (2,5) (3,5) (4,5)
(1,4) (2,4) (3,4) (4,4)
(1,3) (2,3) (3,3) (4,3) …
(1,2) (2,2) (3,2) (4,2)
(1,1) (2,1) (3,1) (4,1)
Another mapping
The set N is equipotent with the set of pairs of
positive integers
Therefore, the set of rational numbers Q is also
equipotent with Z and N, since very rational
number can be represented by a pair of integers
0
Many common infinite sets are equipotent with the
set of integers
This cardinality is written 0
Pronounced “aleph-naught”
A set with this cardinality is said to be countably
infinite because we can put its elements in a 1:1
correspondence with N
Uncountable Infinity
The set R of real numbers is infinite since it contains
all the integers.
Is it countably infinite?
R is equipotent with the set of reals
between 0 and 1
x
1
y
r=1/
0
x=arctan(x)+( /2)+ )
y=tan(y-(/2))
Disproving Equipotency
If you can find a 1:1 correspondence, you have
proven equipotency
If you cannot find a 1:1 correspondence you have
proven nothing, either way.
To disprove equipotency you must prove that no 1:1
correspondence is possible
Alternately, if you prove one infinity countable and
another infinity uncountable you have proven that
the two sets are not equipotent
There is no one-to-one
correspondence between the reals
(0,1) and N
Suppose there was
(some table listing all the real numbers)
We can construct a new number not on the table
The value of the ith digit depends on the ith digit of
the ith real in the table
If said digit is 0-4, new digit is “8”
If said digit is 5-9, new digit is “1”
Diagonalization
If the new real was all ready on the table (the rth),
then it would have contributed the rth digit to the
new number
But the rth digit of the new number differs by at least
4 digits