{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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 , then there there is a °nite number of sets 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 satis°es the 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 E . Then b R = R ° ( E ) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online