Set Theory, Topology from GR9367

Forum for the GRE subject test in mathematics.
CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm

Set Theory, Topology from GR9367

Postby CoCoA » Sun Sep 14, 2008 2:31 pm

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

CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm

Postby CoCoA » Wed Sep 17, 2008 12:18 am

1. I didn't know how to solve this. Is it a simple counting argument? In the Venn Diagram, there are at most 4 "parts" [A-B,AnB,B-A, M-(AuB)}, and the number of possible combinations of these is 2^4? Do I need to be concerned if some combination can be achieved?

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?

amateur
Posts: 42
Joined: Wed Sep 10, 2008 9:41 am

Postby amateur » Wed Sep 17, 2008 7:44 am

I tried Q. 1 as follows:

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?

CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm

Postby CoCoA » Wed Sep 17, 2008 3:34 pm

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.

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!




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