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# Cardinality

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Cardinality (presentation for Math 101 Fall 2008)

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### Cardinality

1. 1. Cardinality Introduction to Analysis December 1, 2008 Samantha Wong
2. 2. Cardinality <ul><li>Cardinality is the number of elements in a set. </li></ul><ul><li>For Example: </li></ul><ul><li>S = {1, 5, 8, 10}. </li></ul><ul><li>Then this set contains four elements. </li></ul>
3. 3. Some Definitions <ul><li>Two sets S and T are called equinumerous if there exists a bijective function from S onto T . We write S~T. </li></ul><ul><li>The cardinal number of a set I n is n , and if S ~ I n , we say that S has n elements. </li></ul>
4. 4. Notation <ul><li>We denote the cardinal number of a set S, as |S| . </li></ul><ul><li>As in the previous example: </li></ul><ul><li>S = {1, 5, 8, 10}. </li></ul><ul><li>Then |S| = 4 </li></ul>
5. 5. Ordinal Numbers <ul><li>An ordinal number tells us the position of an element in a set. </li></ul><ul><li>Going back to our example: </li></ul><ul><li>S = {1, 5, 8, 10}. Then, </li></ul><ul><li>1 is the first ordinal </li></ul><ul><li>5 is the second ordinal </li></ul><ul><li>8 is the third ordinal </li></ul><ul><li>10 is the fourth ordinal. </li></ul>
6. 6. Ordinal Numbers Second Third . . . Two Three . . . First One Ordinal Cardinal
7. 7. Ordinal Numbers <ul><li>Example: </li></ul><ul><li>A = {a, b, c}. a is the first element, b the second, c the third. So, we have three elements, and |A| = 3. </li></ul>
8. 8. Some Definitions <ul><li>Finite : A set S is finite if S is equal to the empty set, or if there exists n an element of the natural numbers, and a bijection f :{1,2,…n}  S. </li></ul><ul><li>Infinite : A set is infinite if it is not finite. </li></ul>
9. 9. Some Definitions (cont’d) <ul><li>Denumerable :A set S is denumerable if there exists a bijection f : N  S. </li></ul><ul><li>Countable : A set is countable if it is finite or denumerable. </li></ul><ul><li>Uncountable : A set is uncountable if it is not countable. </li></ul>
10. 10. A Bit of Cardinal Arithmetic <ul><li>Let s=|S|, and w=|W|. Then: </li></ul><ul><li>s + w=|S|U|W|=|SUW| </li></ul><ul><li>s x w = |S| x |W| = |S x W| </li></ul><ul><li>s w = |S| |W| = |S W | </li></ul>
11. 11. Cardinal Numbers <ul><li>Back to our example: </li></ul><ul><li>S = {1, 5, 8, 10}. |S|=4. </li></ul><ul><li>S is finite, because it has finitely many elements. </li></ul>
12. 12. The Cardinality of Natural Numbers <ul><li>The set of natural numbers is not finite , but it is countable . </li></ul><ul><li>| N | =  0 </li></ul>
13. 13. Example One <ul><li>The cardinality of the natural numbers and even natural numbers is the same. </li></ul><ul><li>Let E = even natural numbers. </li></ul><ul><li>Let N = natural numbers. </li></ul><ul><li>Bijection f : N  E , where f(n)=2n. </li></ul><ul><li>Then E has the same cardinality as N. </li></ul><ul><li>| E | =  0 = | N | </li></ul>
14. 14. Example Two <ul><li>The cardinality of the odd natural numbers and the even natural numbers are the same. </li></ul><ul><li>Let O = odd natural numbers. </li></ul><ul><li>Bijection f : O  E , where f(n) = n+1. </li></ul><ul><li>Then O has the same cardinality as E (and N). </li></ul><ul><li>|O| = |E| =  0 = | N | </li></ul>
15. 15. Example Three <ul><li>E + O = N </li></ul><ul><li>Since we know: </li></ul><ul><li> |E|=  0 , |O|=  0 , | N |=  0 </li></ul><ul><li>Then, |E| + |O| = |N| </li></ul><ul><li>gives us  0 +  0 =  0 . </li></ul>
16. 16. Definition <ul><li>Power set : Given any set S , let P(S) denote the collection of subsets of S . Then P(S) is called the power set of S . </li></ul><ul><li>For example: </li></ul><ul><li>Let S = {1,2}. </li></ul><ul><li>Then, P(S) = {  , {1},{2},{1,2}}. </li></ul><ul><li>*Note that |S| < |P(S)| </li></ul>
17. 17. Theorem <ul><li>For any set S, |S| < |P(S)|. </li></ul>
18. 18. Theorem <ul><li>Any subset of a countable set is countable. </li></ul>
19. 19. The Cardinality of Real Numbers <ul><li>Theorem: </li></ul><ul><li>The set of real numbers is uncountable . </li></ul><ul><li>We denote the cardinality of the real numbers as: </li></ul><ul><li>| R | = C </li></ul>
20. 20. The Real Numbers are Uncountable (Proof) <ul><li>Proving the real numbers are uncountable. </li></ul><ul><li>Assume that R is countable. </li></ul><ul><li>Construct a number that is not in the set. </li></ul><ul><li>By constructing a number not in our original set, we conclude that R is uncountable. </li></ul>
21. 21. The Real Numbers are Uncountable (Proof) <ul><li>Assume that the set of real numbers is countable. </li></ul><ul><li>Then any subset of the real numbers is countable (by the previous theorem). </li></ul><ul><li>So let us look at the set </li></ul><ul><li>S = (0,1) </li></ul>
22. 22. The Real Numbers are Uncountable (Proof) <ul><li>Since we have defined S to be countable, we can list all elements of S . </li></ul><ul><li>So S = { s 1 , s 2 , … , s n } </li></ul>
23. 23. The Real Numbers are Uncountable (Proof) <ul><li>so we can write any element of S in its decimal expansion. Meaning, </li></ul><ul><li>s 1 = 0. a 11 a 12 a 13 a 14 … </li></ul><ul><li>s 2 = 0. a 21 a 22 a 23 a 24 … </li></ul><ul><li>and so on. </li></ul><ul><li>And each a ij is an element of </li></ul><ul><li>{0,1, 2, 3, 4, 5, 6, 7, 8, 9}. </li></ul>
24. 24. The Real Numbers are Uncountable (Proof) <ul><li>Let y = 0. b 1 b 2 b 3 b 4 … </li></ul><ul><li>Where: </li></ul><ul><li>b i = {1, if a nn ≠ 1; 8 if a nn = 1}. </li></ul>
25. 25. The Real Numbers are Uncountable (Proof) <ul><li>For example, if </li></ul><ul><li>x 1 = 0. 3 2045…. </li></ul><ul><li>x 2 = 0.4 4 246… </li></ul><ul><li>x 3 = 0.57 1 24… </li></ul><ul><li>Then y = 0. 1 1 8 … </li></ul>
26. 26. The Real Numbers are Uncountable (Proof) <ul><li>y is made up of 1’s and 8’s, so y is in S = (1,0) </li></ul><ul><li>But, y ≠ s n because it differs from s n at the nth decimal place. </li></ul><ul><li>S must be uncountable. </li></ul><ul><li>Then the real numbers are uncountable. </li></ul>
27. 27. Recall… <ul><li> Since the real numbers are uncountable, and the natural numbers are countable: </li></ul><ul><li>|N| < |R| </li></ul><ul><li> 0 < C </li></ul><ul><li>There are more real numbers than natural numbers! </li></ul>
28. 28. Hmm… <ul><li> 0 < ? < C </li></ul>
29. 29. The Continuum Hypothesis <ul><li>Cantor believed his sequence, </li></ul><ul><li>0, 1, 2, …,  0 ,  1 ,  2 , …,   </li></ul><ul><li>contained every cardinal number. </li></ul><ul><li>But, which one is C? </li></ul><ul><li> 0 is the number of finite ordinal numbers. </li></ul><ul><li> 1 is the number of ordinal numbers that are either finite or in the  0 class. </li></ul><ul><li>And so on… </li></ul>
30. 30. The Continuum Hypothesis (cont’d) <ul><li>There are exactly C = 2  0 real numbers and C >  0 . </li></ul><ul><li>But, does C =  1 ? </li></ul><ul><li>Cantor believed so. </li></ul>
31. 31. The Generalized Continuum Hypothesis <ul><li> α +1 = 2  α </li></ul><ul><li>for all α ? </li></ul>
32. 32. The Continuum Hypothesis <ul><li> 0 < ? < C </li></ul>l Georg Cantor suggested that no such set exists . Kurt Godel showed that this couldn’t be disproved . Paul Cohen showed that this couldn’t be proved either. 1900 1940 1963
33. 33. End