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Danielle Bassignani Duncan Berube Mentor: Tony Julianelle Math052B Team Project

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.".

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Danielle Bassignani Duncan Berube Mentor: Tony Julianelle Math052B Team Project

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  1. Cardinality, eh? I wonder if I’m related to that. Danielle Bassignani Duncan Berube Mentor: Tony Julianelle Math052B Team Project

  2. 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

  3. What is cardinality? The cardinality of a set S, denoted |S|, is the number of elements in S.

  4. For the usual finite sets you can think of finding the cardinality is simply counting the elements in the set.

  5. 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.

  6. 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.

  7. Words representing the rank of a number with respect to some order, in particular order or position Ordinal numbers

  8. 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)]

  9. 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.

  10. 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, 4, 5, ... n, ... ... ... 1, 4, 9, 16, 25, ... 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.

  11. Each Lion is assigned a number. Each lion gets a friend sheep. Therefore they are one-to-one and onto. Two sets L and S have the same cardinality if there is a bijection f:L -> S.

  12. Even though N, Z and R are all infinite sets, their cardinalities are not all the same.

  13. "Counting Numbers and Rational Numbers." CooperToons. Web. 21 Apr. 2012. <http://www.coopertoons.com/education/countingrationals/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. References

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