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

## Set Theory, Topology from GR9367

2. item I: true: if x in (AuB)', then x in cl(AuB-{x}), so x in cl(A-{x}) u cl(B-{x}) because cl(AuB-{x}) is the smallest closed set containing AuB-{x}, thus x in A'uB'. if x in A'uB', then assume that x in A', so x in cl(A-{x}), thus x in cl(AuB-{x}), therefore x in (AuB)'.

item III: true: because then cl(A)=AuA'=A so A is closed.

item IV: false: if X has the discrete topology then there are NO limit points and every subset is open.

item II is apparently false, but what is an easy counterexample? Does A=[0,1), B=[1,2) in the lower limit topology work?

S0 = {A, B}

S1 = {A U A, A U B, B U A, B U B, A n A, A n B, B n A, B n B, M - A, M - B} which simplifies to

S1 = {A, B, A U B, A n B, A', B'}

S2 = {A U A, A U B, A U (A U B), A U (A n B), A U A', A U B', B U A, B U B, B U (A U B), B U (A n B), B U A', B U B', A n A, ...}

I couldn't simplify S2.

As I understood it, the question asks if S(infinity) will have a finite number of elements or not; and if so how many?

My understanding also.As I understood it, the question asks if S(infinity) will have a finite number of elements or not; and if so how many?

From the Venn Diagram, we will never have a set in S_n that contains only a part of A-B, but not all of A-B. Same for AnB, B-A, and M-(AuB). So the most sets that can be in S_\infty is 2^4 = 16.

Then instead of figuring out each S_n, just ensure that each of the 16 possibilities is in some S_n. (I don't know if this is really necessary, or if there is an easy way to know that they all must be in some S_n.)

1. A in S_1

2. B in S_1

3. AuB in S_1

4. AnB in S_1

5. M-A in S_1

6. M-B in S_1

7. M-(AuB) in S_2

8. M-(AnB) in S_2

9. A-B in S_2 [ it's (M-B)nA ]

10. B-A in S_2

11. [M-(AuB)]u(AnB) in S_3

12. (A-B)u(B-A) in S_3

13. M-(B-A) in S_3

14. M-(A-B) in S_3

15. M in S_2

16. null in S_2

I thought of none of this while doing the practice exam!