INFIINITE SETS

CSC 172

SPRING 2002

LECTURE 23

Cardinality and CountingThe cardinality of a set is the number of elements in

the setTwo sets are equipotent if and oly if they have the

same cardinalityThe existence of a one-to-one correspondence

between two sets proves that they are equipotentCounting 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 setsPossible with infinite setsTechnically, 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 evenThe mapping goes both ways, so Z is equipotent with N

…4,3,2,{1

{…,-2,-1,0,1,2,..}

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

r=1/

0

1 x=arctan(x)+( /2)+ )y=tan(y-(/2))

x

y

Disproving EquipotencyIf 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 NSuppose 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