(a) Let X be a set and B₁, B2, ..., Bn be a partition of X, for some n E N. Let A be the collection of subsets of X consisting of the empty set and finite unions of the Bi's. Show that A is a o-algebra on X. 5) Let (Ca)acl be a collection of monoton classes on a nonemty set X. Show that the intersection of the collection is a monotone class. Show that every o-algebra on X is a monotone class.

Let T be the set of real numbers. Consider T as an algebraic structure with addition operation and multiplication operation given by xy = min{x,y}, X y = x + y - 2. Name the identity element in T for the operation, and prove the inverse law for O. 2 Prove the distributive law in T. [Hint: consider two cases x ≤ y, x > y.] 3 Prove that the set T with addition © and multiplication © is not a ring.

Let {An : n e N} be the indexed collection of sets defined by (2. 2-) 1 An Prove that: = Ø

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