540hw4 - HW4 1. Let X be an uncountable set and R X be the...

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HW4 1. Let X be an uncountable set and R X be the family of all sets which either countable or their complements (in X ) are countable. Prove that R is a ring. 2. Describe the minimal ring generated by F in the following cases: Describe the minimal ring generated by F in the following cases: (i) Let F be a family consisting of only one non-empty set E : F = f E g . (ii) Let F be the family consisting of all sets which contain exactly two points. 3. Let H be a family of sets such that (I) A;B 2 H ) A \ B 2 H : (II) If A;B 2 H and A B C 0 ;C 1 ;:::C n 1 ;C n such that (a) C 0 = A C 1 C 2 ::: C n 1 C n = B . (b) C j n C j 1 2 H for every j = 1 ; 2 ;:::;n . Prove that a family of sets is a semiring if and only if H properties (I) and (II). 4. (a) Let E be a family of sets and R ( E ) be the minimal & -ring gen- erated by E . Let b R be the union of all minimal & -rings generated by countable subfamilies of sets of the family of sets
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540hw4 - HW4 1. Let X be an uncountable set and R X be the...

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