Problem 1. Let X be a nonempty set. A partition of X is a finite collection of non-empty disjoint subsets of X, P = {X, X, ..., X}, such that X = X UX U... U Xk. We say that the positive integer k is the length of the partition P. For example, a partition of R is P = {(-,0), {0}, (0, )}. It is not difficult to see that R has an infinite number of different partitions. If we instead consider a set of only two elements, X = {, 2}, then there are only two different partitions of X, namely P = {{1}, {2}} and P = {{x, x}}. (a) Find all different partitions of the sets X = {1, 2, 3} and Y = {y1, y2, 93, Y4}. Hint. OEIS A000110. (b) Let P = {X, X2,..., Xk} be a partition of a nonempty set X. Let A = o(P) be the o-algebra generated by P. Prove that every A in A can be expressed as A = UX jEJ for some subset J of {1,2,..., k}. What is the number of sets in the o-algebra o(P)? (c) Show that if P and P are different, then o(P) and o(P) are different. A o-algebra A is called finite if A contains a finite number of sets. If A is finite, we let A denote the number of sets in A. (d) Prove that if X is a non-empty set and A is a finite o-algebra on X, then there is a partition P such that A = o(P). = (e) Consider the set Z {21, 22, 23, 24, 25}. Prove that there are 52 different o-algebras on Z. For which integers n is there a o-algebra A on Z such that |A| = n? Problem 2. Let R consist of the following subsets of R: (i) . (ii) All finite, half-open intervals (a, b), where a, b E R and a < b.