1. Let A and B be subsets of a set M and let S_0={A,B}. For i>=0, define S_{i+1} inductively to be the collection of subsets X of M that are of the form CuD, CnD or M-C, where C,D in S_i. Let S= union_{i=0}^{\infty}{S_i}. What is the largest possible number of elements of S?

A. 4

B. 8

C. 15

D. 16

E. S may be infinite.

2. For a subset S of a topological space X, let cl(S) denote the closure of S in X, and let S'={x:x in cl(S-{x})} denote the derived set of S. If A and B are subsets of X, which of the following statements are true?

I. (AuB)'=A'uB'

II. (AnB)'=A'nB'

III. If A' is empty, then A is closed in X.

IV. If A is open in X, then A' is not empty.

A. I and II only

B. I and III only

C. II and IV only

D. 1, II, and III only

E. I, II, III, and IV