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Unformatted text preview: Administration 1. Midterm regrades due today! 2. Homework due on Friday. CS70: Satish Rao: Lecture 37. I Countability. I Uncountability. Cardinality. A bijection is function f : S → T , Cardinality. A bijection is function f : S → T , I onetoone, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) Cardinality. A bijection is function f : S → T , I onetoone, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Cardinality. A bijection is function f : S → T , I onetoone, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Cardinality. A bijection is function f : S → T , I onetoone, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Two sets, S and T have the same cardinality if there is a bijection from S to T Cardinality. A bijection is function f : S → T , I onetoone, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Two sets, S and T have the same cardinality if there is a bijection from S to T If there is a bijection from S to T , there is one from T to S Countable. Countable. Definition: S is countable if there is a bijection between S and some subset of N . Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Bijection to or from natural numbers implies countably infinite. Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Bijection to or from natural numbers implies countably infinite. Enumerable means countable. Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Bijection to or from natural numbers implies countably infinite. Enumerable means countable. Subset of countable set is countable. Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality ....
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 Fall '11
 Rau
 Natural number, Countable set, reals

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