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chap2-3slidespt2

chap2-3slidespt2 - Set theory Set algebra Example Let S =...

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Set theory Set algebra Example: Let S = { a, b } with P S = {∅ , { a } , { b } , { a, b }} { a } is a subset of S that contains a single element ( a ) of S ; { a } ∈ P S (avoid writing a ∈ P S ) ECSE 305 - Winter 2012 40
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Set theory Set algebra Definition A set F of subsets of S , that is, F ⊆ P S , is an algebra iff 1 S ∈ F 2 A ∈ F ⇒ A c ∈ F 3 A, B ∈ F ⇒ A B ∈ F ∅ ∈ F since = S c . The algebra F is closed under the operations of complementation and union. ECSE 305 - Winter 2012 41
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Set theory Set algebra Example: Let S = { a, b } with P S = {∅ , { a } , { b } , { a, b }} ,and F = {∅ , { a } , { a, b }} F ⊂ P S { b } 6∈ F . { b } ⊂ { a, b } ∈ F 6⇒ { b } ∈ F . Is F an algebra? ECSE 305 - Winter 2012 42
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Set theory Set algebra Example (contd): F = {∅ , { a } , { a, b }} 1 S = { a, b } ∈ F ? YES 2 A ∈ F ⇒ A c ∈ F ? NO { a } c = { b } 6∈ F 3 A, B ∈ F ⇒ A B ∈ F ? YES { a } ∪ { a, b } = { a, b } ∈ F ECSE 305 - Winter 2012 43
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Set theory Set algebra An algebra F is also closed under the operation of intersection, that is: A, B ∈ F ⇒ A B ∈ F . Indeed, we have A, B ∈ F A c , B c ∈ F A c B c ∈ F ( A c B c ) c ∈ F A B ∈ F ECSE 305 - Winter 2012 44
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Set theory σ -algebra Definition A set F of subsets of S , that is, F ⊆ P S , is a σ -algebra iff 1 S ∈ F 2 A ∈ F ⇒ A c ∈ F 3 If the sets A 1 , A 2 , A 3 , ... belong to F , so does i =1 A i . The algebra F is closed under under any countable combination of complement, union and intersection operations. Any finite algebra (i.e. F contains a finite number of elements) is also a σ -algebra. ECSE 305 - Winter 2012 45
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Set theory R is uncountable (simplified proof) It suffices to show that A = [0 , 1] ⊂ R is uncountable. Suppose that A is countable: A = { r 1 , r 2 , . . . , r n , . . . } ECSE 305 - Winter 2012 46
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Set theory R is uncountable (simplified proof) Decimal expansion: r 1 = 0 .d 11 d 12 d 13 . . . d 1 n . . . r 2 = 0 . d 21 d 22 d 23 . . . d 2 n . . . . . . r n = 0 . d n 1 d n 2 d n 3 . . . d nn . . . . . . ECSE 305 - Winter 2012 47
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Set theory R is uncountable (simplified proof) Decimal expansion: r 1 = 0 . d 11 d 12 d 13 . . . d 1 n . . . r 2 = 0 . d 21 d 22 d 23 . . . d 2 n . . . . . . r n = 0 . d n 1 d n 2 d n 3 . . . d nn . . . . . . x = 0 .d 11 d 22 d 33 . . . d nn . . . ECSE 305 - Winter 2012 48
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Set theory R is uncountable (simplified proof) Decimal expansion: r 1 = 0 .
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