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Cardinality Version 2
 

Cardinality Version 2

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Cardinality presentation version 2 for Math 101 Fall 2008

Cardinality presentation version 2 for Math 101 Fall 2008

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Cardinality Version 2 Cardinality Version 2 Presentation Transcript

  • Cardinality Introduction to Analysis December 1, 2008 Samantha Wong
  • Cardinality
    • Cardinality is the number of elements in a set.
    • For Example:
    • S = {1, 5, 8, 10}.
    • Then this set contains four elements.
  • Some Definitions
    • Two sets S and T are called equinumerous if there exists a bijective function from S onto T . We write S~T.
    • The cardinal number of a set I n is n , and if S ~ I n , we say that S has n elements.
  • 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
  • Ordinal Numbers
    • An ordinal number tells 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.
  • Ordinal Numbers Second Third . . . Two Three . . . First One Ordinal Cardinal
  • 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.
  • Some Definitions
    • 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.
    • Infinite : A set is infinite if it is not finite.
  • Some Definitions (cont’d)
    • Denumerable :A set S is denumerable if there exists a bijection f : N  S.
    • Countable : A set is countable if it is finite or denumerable.
    • Uncountable : A set is uncountable if it is not countable.
  • 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 |
  • Cardinal Numbers
    • Back to our example:
    • S = {1, 5, 8, 10}. |S|=4.
    • S is finite, because it has finitely many elements.
  • The Cardinality of Natural Numbers
    • The set of natural numbers is not finite , but it is countable .
    • | N | =  0
  • Example One
    • The cardinality of the natural numbers and even natural numbers is the same.
    • Let E = even natural numbers.
    • Let N = natural numbers.
    • Bijection f : N  E , where f(n)=2n.
    • Then E has the same cardinality as N.
    • | E | =  0 = | N |
  • Example Two
    • The cardinality of the odd natural numbers and the even natural numbers are the same.
    • Let O = odd natural numbers.
    • Bijection f : O  E , where f(n) = n+1.
    • Then O has the same cardinality as E (and N).
    • |O| = |E| =  0 = | N |
  • Example Three
    • E + O = N
    • Since we know:
    • |E|=  0 , |O|=  0 , | N |=  0
    • Then, |E| + |O| = |N|
    • gives us  0 +  0 =  0 .
  • Definition
    • 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 .
    • For example:
    • Let S = {1,2}.
    • Then, P(S) = {  , {1},{2},{1,2}}.
    • *Note that |S| < |P(S)|
  • Theorem
    • For any set S, |S| < |P(S)|.
  • Theorem
    • Any subset of a countable set is countable.
  • The Cardinality of Real Numbers
    • Theorem:
    • The set of real numbers is uncountable .
    • We denote the cardinality of the real numbers as:
    • | R | = C
  • The Real Numbers are Uncountable (Proof)
    • Proving the real numbers are uncountable.
    • Assume that R is countable.
    • Construct a number that is not in the set.
    • By constructing a number not in our original set, we conclude that R is uncountable.
  • 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)
  • The Real Numbers are Uncountable (Proof)
    • Since we have defined S to be countable, we can list all elements of S .
    • So S = { s 1 , s 2 , … , s n }
  • The Real Numbers are Uncountable (Proof)
    • so we can write any element of S in 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 each a ij is an element of
    • {0,1, 2, 3, 4, 5, 6, 7, 8, 9}.
  • The Real Numbers are Uncountable (Proof)
    • Let y = 0. b 1 b 2 b 3 b 4 …
    • Where:
    • b i = {1, if a nn ≠ 1; 8 if a nn = 1}.
  • 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…
    • Then y = 0. 1 1 8 …
  • The Real Numbers are Uncountable (Proof)
    • y is made up of 1’s and 8’s, so y is in S = (1,0)
    • But, y ≠ s n because it differs from s n at the nth decimal place.
    • S must be uncountable.
    • Then the real numbers are uncountable.
  • 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!
  • Hmm…
    •  0 < ? < C
  • The Continuum Hypothesis
    • Cantor believed his sequence,
    • 0, 1, 2, …,  0 ,  1 ,  2 , …,  
    • contained every cardinal number.
    • But, which one is C?
    •  0 is the number of finite ordinal numbers.
    •  1 is the number of ordinal numbers that are either finite or in the  0 class.
    • And so on…
  • The Continuum Hypothesis (cont’d)
    • There are exactly C = 2  0 real numbers and C >  0 .
    • But, does C =  1 ?
    • Cantor believed so.
  • The Generalized Continuum Hypothesis
    •  α +1 = 2  α
    • for all α ?
  • The Continuum Hypothesis
    •  0 < ? < C
    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
  • End