chap2-3slidespt2

# Chap2-3slidespt2 - Set theory Set algebra Example Let S = cfw_a b with PS = cfw cfw_a cfw_b cfw_a b cfw_a is a subset of S that contains a single

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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
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
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
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
Set theory R is uncountable (simpliﬁed proof) Decimal expansion: r 1 = 0 . d

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## This note was uploaded on 02/12/2012 for the course ECSE 305 taught by Professor Champagne during the Spring '09 term at McGill.

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Chap2-3slidespt2 - Set theory Set algebra Example Let S = cfw_a b with PS = cfw cfw_a cfw_b cfw_a b cfw_a is a subset of S that contains a single

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