Transcript Slide 1
Cardinality, eh?
I wonder if I’m
related
to that.
Danielle Bassignani Duncan Berube
Mentor: Tony Julianelle
Math052B Team Project
GEORG CANTOR
German mathematician who has
gone down in history as the
founder of set theory.
"I see it but I don't believe it."
There is no one-to-one matching
between N and R.
Multiple Infinities
WHAT IS CARDINALITY?
The cardinality of a set S, denoted |S|, is the number of
elements in S.
For the usual finite sets you can think
of finding the cardinality is simply
counting the elements in the set.
But this will not always work.
How could counting “not
work”?
It breaks down when we
consider
infinite sets.
*** If you’re not Chuck Norris,
you can’t just count to infinity.
CARDINAL NUMBERS
These are known as counting numbers, they show a
quantity. This is the set of all natural numbers which is
infinite, countably infinite.
The cardinality of the set of natural numbers
N = {0, 1, 2, 3, ...} is
“aleph-zero”
This is a transfinite number,
not equal to any finite number.
ORDINAL NUMBERS
Words
representing the
rank of a number
with respect to
some order, in
particular order or
position
How do we know this set of numbers is
infinite? Well. . . no matter what number you
pick, you can always add one to the number
and get another number, the definition of
infinite.
[ Mathematical Induction P(N) -> P(N+1)]
Any set whose elements could be put into a
one-to-one correspondence with counting
numbers is also countably infinite. This
correspondence is a simple extension of how
we know finite sets are the same size.
Galileo’s Paradox
There are as many perfect squares as there
are natural numbers. This can be seen by
pairing the natural numbers with the perfect
squares to show that there is a one-to-one
correspondence between the two sets:
1,
2,
3,
1,
4,
9,
4,
5,
...
...
16, 25, ...
n,
n2
...
...
...
it seems evident that most natural numbers
are not perfect squares, so that the set of
perfect squares is smaller than the set of all
natural numbers.
Each Lion is assigned a
number. Each lion gets
a friend sheep.
Therefore they are oneto-one and onto.
Two sets L and S have the same cardinality if there is a bijection f:L -> S.
Even though N, Z and R are all infinite sets,
their cardinalities are not all the same.
REFERENCES
"Counting Numbers and Rational
Numbers." CooperToons. Web. 21 Apr. 2012.
<http://www.coopertoons.com/education/counti
ngrationals/cantorsrationalnumbers.html>.
Epp, Susanna S. "Cardinality with Applications to
Computability." Discrete Mathematics with
Applications. 2nd ed. Boston: PWS Pub., 1995.
411-22. Print.
Velleman, Daniel J. "Infinite Sets." How to Prove It:
A Structured Approach. Cambridge: Cambridge UP,
2006. 306-28. Print.