# cardinality version 2

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Cardinality presentation version 2 for Math 101 Fall 2008TRANSCRIPT

- 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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