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# Cardinality Version 2

## on Nov 30, 2008

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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 2Presentation 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