cardinality version 2

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Cardinality presentation version 2 for Math 101 Fall 2008

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  • 1. Cardinality Introduction to Analysis December 1, 2008 Samantha Wong

2. Cardinality

  • Cardinality is the number of elements in a set.
  • For Example:
  • S = {1, 5, 8, 10}.
  • Then this set contains four elements.

3. Some Definitions

  • Two setsSandTare calledequinumerousif there exists a bijective function fromSontoT . We write S~T.
  • Thecardinal numberof a setI nisn , and ifS ~ I n ,we say thatShasnelements.

4. Notation

  • We denote the cardinal number of a set S, as|S| .
  • As in the previous example:
  • S = {1, 5, 8, 10}.
  • Then |S| = 4

5. Ordinal Numbers

  • Anordinal numbertells us the position of an element in a set.
  • Going back to our example:
  • S = {1, 5, 8, 10}. Then,
  • 1 is the first ordinal
  • 5 is the second ordinal
  • 8 is the third ordinal
  • 10 is the fourth ordinal.

6. Ordinal Numbers Second Third . . . Two Three . . . First One Ordinal Cardinal 7. Ordinal Numbers

  • Example:
  • A = {a, b, c}. a is the first element, b the second, c the third. So, we have three elements, and |A| = 3.

8. Some Definitions

  • Finite : A setSis finite ifSis equal to the empty set, or if there existsnan element of the natural numbers, and a bijectionf :{1,2,n}S.
  • Infinite : A set is infinite if it is not finite.

9. Some Definitions (contd)

  • Denumerable :A set S is denumerable if there exists a bijectionf : N S.
  • Countable : A set is countable if it is finite or denumerable.
  • Uncountable : A set is uncountable if it is not countable.

10. A Bit of Cardinal Arithmetic

  • Let s=|S|, and w=|W|. Then:
  • s + w=|S|U|W|=|SUW|
  • s x w = |S| x |W| = |S x W|
  • s w = |S| |W|= |S W |

11. Cardinal Numbers

  • Back to our example:
  • S = {1, 5, 8, 10}. |S|=4.
  • S is finite, because it has finitely many elements.

12. The Cardinality of Natural Numbers

  • The set ofnatural numbersisnot finite , but itis countable .
  • | N | = 0

13. Example One

  • The cardinality of the natural numbers and even natural numbers is the same.
  • Let E = even natural numbers.
  • LetN= natural numbers.
  • Bijectionf : N E , where f(n)=2n.
  • ThenEhas the same cardinality asN.
  • | E | = 0= | N |

14. Example Two

  • The cardinality of the odd natural numbers and the even natural numbers are the same.
  • LetO= odd natural numbers.
  • Bijectionf :O E , where f(n) = n+1.
  • ThenOhas the same cardinality asE (and N).
  • |O| = |E| = 0= | N |

15. Example Three

  • E+O= N
  • Since we know:
  • |E|= 0 ,|O|= 0,| N |= 0
  • Then,|E| + |O| = |N|
  • gives us 0+ 0= 0 .

16. Definition

  • Power set : Given any setS , letP(S)denote the collection of subsets ofS . ThenP(S)is called the power set ofS .
  • For example:
  • Let S = {1,2}.
  • Then, P(S) = { , {1},{2},{1,2}}.
  • *Note that |S| < |P(S)|

17. Theorem

  • For any set S, |S| < |P(S)|.

18. Theorem

  • Any subset of a countable set is countable.

19. The Cardinality of Real Numbers

  • Theorem:
  • The set ofreal numbersisuncountable .
  • We denote the cardinality of the real numbers as:
  • | R | = C

20. The Real Numbers are Uncountable (Proof)

  • Proving the real numbers are uncountable.
  • Assume thatRis countable.
  • Construct a number that is not in the set.
  • By constructing a number not in our original set, we conclude thatRis uncountable.

21. The Real Numbers are Uncountable (Proof)

  • Assume that the set of real numbers is countable.
  • Then any subset of the real numbers is countable (by the previous theorem).
  • So let us look at the set
  • S= (0,1)

22. The Real Numbers are Uncountable (Proof)

  • Since we have definedSto be countable, we can list all elements ofS .
  • SoS= { s 1 ,s 2 , ,s n }

23. The Real Numbers are Uncountable (Proof)

  • so we can write any element ofSin its decimal expansion. Meaning,
  • s 1= 0. a 11 a 12 a 13 a 14
  • s 2= 0. a 21 a 22 a 23 a 24
  • and so on.
  • And eacha ij is an element of
  • {0,1, 2, 3, 4, 5, 6, 7, 8, 9}.

24. The Real Numbers are Uncountable (Proof)

  • Lety=0. b 1 b 2 b 3 b 4
  • Where:
  • b i= {1, ifa nn 1; 8 ifa nn= 1}.

25. The Real Numbers are Uncountable (Proof)

  • For example, if
  • x 1 = 0. 3 2045.
  • x 2 = 0.4 4 246
  • x 3= 0.57 1 24
  • Theny= 0. 1 1 8

26. The Real Numbers are Uncountable (Proof)

  • yis made up of 1s and 8s, soyis in S = (1,0)
  • But,ys nbecause it differs froms nat the nth decimal place.
  • Smust be uncountable.
  • Then the real numbers are uncountable.

27. Recall

  • Since the real numbers are uncountable, and the natural numbers are countable:
  • |N| < |R|
  • 0< C
  • There are more real numbers than natural numbers!

28. Hmm

  • 0

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