ans5 - ECON 831 Answers Homework #5 Exercise 1: (i) We...

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ECON 831 Answers Homework #5 Exercise 1: (i) We prove: If F is an algebra, then (2'): A , B ∈ F ⇒ ( A B ) ∈ F . Suppose F is an algebra. Pick any A and B ∈ F . Then we have: A,B ∈ F ⇒ A c ,B c ∈ F by property (1) ( A c B c ) ∈ F by property (2) ( A c B c ) c ∈ F ( A B ) ∈ F (ii) We prove: A collection F of subsets of a non-empty set Ω is an algebra if it satis es: (1) A ∈ F ⇒ A c ∈ F and (2'). Suppose now that properties (1) and (2') are satis ed. We need to show that F is an algebra. It remains to show that (2) is satis ed: A,B ∈ F ⇒ A c ,B c ∈ F ( A c B c ) ∈ F by property (2') ( A c B c ) c ∈ F by property (1) ( A B ) ∈ F property (1) is satis ed ⇒ F is an algebra Exercise 2: We need to verify properties (1) and (3) for F = T θ Θ F θ . 1
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- Pick any A ∈ F , then we have: A ∈ F ⇒ A ∈ F θ θ Θ A c ∈ F ∀ θ Θ (property (1) applied to each algebra F θ ) A
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ans5 - ECON 831 Answers Homework #5 Exercise 1: (i) We...

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